Grade 12 / Primitives and differential equations 58 exercises (including 48 corrected)

a
Cϕ Cf Cg Ch -4-3-2-1234I-4-3-2-1234JO -4-3-2-1234I-4-3-2-1234JOCf 1. Introduction to primitives E.3575 Consider four functions ’ , f , g , h de-fined on the interval − 4 ; 4 . Their representative curves are given, below, in the same orthonormal reference frame : Which of f , g and h can admit the function ’ as a derived function? E.3560 Let f be a function derivable on the interval − 4 ; 4 and admitting the function f as derivative. In the reference frame below, is drawn the representative curve of the function f . 1 a What is the value of the number derived from the function f in 1 ? b Give the slope of all tangents represented on the line with equation x =1 . 2 Draw a representation ˇ possible ı of the function f in this frame of reference. E.3561 Consider a function f derivable on − 4 ; 4 . In the orthonormal frame of reference O ; I ; J below is given the representative curve C f of the function f derived from the function f . 1 a What is the derivative number of the function f in − 2 . b All along the straight line of equation x = − 2 are rep-resented tangents ; what is the coefficient of these tan-gents. 2 Draw a representative curve C f acceptable of the func-tion f whose tangents at points of abscissa − 4 , − 2 , 0 , 2 , 4 are, in each case, one of the tangents proposed on the graph. 2. Differential equation: y = f E.6911 Complete the blanks : 1 We note f a function verifying: f ( x )=2 · x . A possible expression of f is : f ( x ) = : : : : : : : : : : : : : : : : : : 2 We note g a function verifying: g ( x )= x 2 . A possible expression of g is : g ( x ) = : : : : : : : : : : : : : : : : : : 3 We note h a function verifying: h ( x )= − 2 . A possible expression of h is : h ( x ) = : : : : : : : : : : : : : : : : : : 4 We note j a function verifying: j ( x )= 1 x 2 . A possible expression of j is : https://chingmath.fr chapExoCorrec/3575 sacados/3575 Cϕ Cf Cg Ch chapExoCorrec/3560 sacados/3560 -4-3-2-1234I-4-3-2-1234JO chapExoCorrec/3561 sacados/3561 -4-3-2-1234I-4-3-2-1234JOCf chapExoCorrec/6911 sacados/6911
j ( x ) = : : : : : : : : : : : : : : : : : : 5 We note k a function verifying: k ( x )= − 1 x . A possible expression of k is : k ( x ) = : : : : : : : : : : : : : : : : : : 6 We note ‘ a function verifying: ‘ ( x )=e x . A possible expression of ‘ is : ‘ ( x ) = : : : : : : : : : : : : : : : : : : 7 Let m be a function verifying: m ( x )= 1 x . A possible expression of m is : m ( x ) = : : : : : : : : : : : : : : : : : : E.5206 For each question, determine the ex-pression of a function f admitting as derivative the proposed expression : a f ( x ) = 3 b f ( x ) = 2 x + 1 c f ( x ) = x 3 d f ( x ) = − 2 x e f ( x ) = 1 √ x f f ( x ) = e 2 x E.6912 Give a primitive of each of the functions below : a f ( x ) = 2 · x + 1 b g ( x ) = 1 − 3 x c h ( x ) = 1 x + 1 x 2 d j ( x ) = e 2 · x E.4292 Consider the function f defined on 0 ; + ∞ by: f ( x ) = ln( x ) + 1 − 1 x 1 Determine the limits of the function f at the bounds of its defining set. 2 Study the variations of the function f on the interval 0 ; + ∞ . 3 Determine the image of 1 by the function f . Deduce the sign of f ( x ) when x describes the interval 0 ; + ∞ . 4 Show that the function F defined on the interval 0 ; + ∞ by: F ( x ) = x · ln x − ln x is a primitive of the function f on this interval. 5 Show that the function F is strictly increasing on the interval 1 ; + ∞ . 6 Show that the equation F ( x )=1 − 1 e admits a single so-lution in the interval 1 ; + ∞ which will be denoted ¸ . 7 Give a frame of ¸ of amplitude 10 − 1 . E.4301 Let f be the function defined on R by: f ( x )= 4 1+7 · e − x 1 Verify that for any real x : f ( x )= 4 · e x e x +7 2 Determine a primitive of the function f on R . 3 Calculate the mean value of f on the interval 0 ; ln7 . 3. Determining a primitive of a reference function E.5207 Determine a primitive of each of the following functions : a f ( x ) = 2 x + 1 b g ( x ) = 1 − 3 x c h ( x ) = 2 x 2 d i ( x )= x 2 + x +1 e j ( x ) = 4 x 3 f k ( x ) = 1 − 2 x 2 E.3992 Determine a primitive of each of the following functions : a f ( x ) = − 1 x 2 b g ( x ) = 2 x 2 c h ( x ) = 1 2 · √ x d j ( x ) = 2 √ x e k ( x ) = 1 x f ‘ ( x ) = − 1 2 x g m ( x ) = e x h n ( x ) = 3e x i p ( x ) = − e x 4. Determining a primitive of the composite of functions E.5209 Determine a primitive of each of the following functions : a f ( x ) = ( x + 3) 4 b g ( x ) = (2 − x ) 3 c h ( x ) = (2 x − 3) 2 d j ( x ) = x · ( x 2 + 1) 6 e k ( x ) = 3 x 2 · ( x 3 − 2) 3 f ‘ ( x ) = x 4 · (1 − x 5 ) 2 E.3930 Determine a primitive for each of the following functions : a f ( x ) = − 3 · x 2 + 2 · x − 1 b g ( x ) = 3 · x 5 + 1 c h ( x ) = (3 + x ) 2 d j ( x ) = (2 − x ) 3 e k ( x ) = (5 x + 1) 4 f ‘ ( x ) = x 3 · x 4 + 1 4 https://chingmath.fr chapExoCorrec/5206 sacados/5206 chapExoCorrec/6912 sacados/6912 chapExoCorrec/4292 sacados/4292 chapExoCorrec/4301 sacados/4301 chapExoCorrec/5207 sacados/5207 chapExoCorrec/3992 sacados/3992 chapExoCorrec/5209 sacados/5209 chapExoCorrec/3930 sacados/3930
E.5214 Determine a primitive of each of the following functions : a f ( x ) = x 2 · ( x 3 − 5) 5 b g ( x ) = x x 2 + 1 c h ( x ) = 6 x + 2 (3 x 2 + 2 x − 5) 2 d j ( x ) = 6 x + 2 3 x 2 + 2 x − 5 E.5211 Determine a primitive of the following functions : a f ( x ) = e 3 x +1 b g ( x ) = x · e x 2 c h ( x ) = e 1 x x 2 d j ( x ) = e √ x x e k ( x ) = e x +1 x x 2 f ‘ ( x ) = e ln( x )+1 x E.5221 Determine a primitive of each of the following functions : a f ( x ) = 3 x − 5 x 5 b g ( x ) = 1 x − x c h ( x ) = x · 2 x 2 − 3 4 d j ( x ) = 2 − x x 2 − 4 x e k ( x ) = 6 x + 2 3 x 2 + 2 x − 3 f ‘ ( x ) = x · e x 2 E.3931 Determine a primitive for each of the following functions : a f ( x ) = x + 1 + 1 x + 1 x 2 b g ( x ) = 1 x + 1 2 c h ( x ) = 1 x + 1 d j ( x ) = x x 2 + 1 2 e ‘ ( x ) = x x + 1 f m ( x ) = 6 x + 1 3 · x 2 + x − 5 2 g n ( x ) = − 3 (3 x + 2) 2 h p ( x ) = 6 x + 1 6 · x 2 + 2 · x + 2 E.5210 Determine a primitive of the following functions : a f ( x ) = 2 2 x + 3 b g ( x ) = 1 1 − 3 x c h ( x ) = x x 2 + 1 d j ( x ) = − 1 (1 + x ) 2 e k ( x ) = 2 (3 x + 1) 2 f ‘ ( x ) = x x 4 + 2 x 2 + 1 E.10393 Let f be a continuous function on an interval I . We define the function g on the interalle I by: g ( x ) = f 3 · x for x ∈ I Let F and G be the primitives respectively of the functions f and g on the interval I . Which of the relationships below is verified by the functions F and G : a G ( x ) = F ( x ) + 3 b G ( x ) = 1 3 · F ( x ) c G ( x ) = F ( x ) − 3 d G ( x ) = 3 · F ( x ) 5. Some special primitives E.6006 1 Consider the function f defined on R by the relation: f ( x ) = x − 1 · e x Determine the expression of the function f derived from the function f . 2 Deduce the expression of a primitive of the function g defined on R by: g ( x ) = x · e x E.6005 1 Consider the function f defined on R + by the expression : f ( x ) = 2 3 · x · √ x Determine the expression of the derivative f of the func-tion f . 2 Deduce the expression of a primitive of the square root function. E.5212 Consider the function f defined by: f ( x ) = x · ln x − x 1 Determine the expression of the derivative function f . 2 Deduce the expression of the primitives of the neperian logarithm function. 6. Finding a primitive E.5222 Let g be the function defined on the interval 1 ; + ∞ by: g ( x ) = 1 x ( x 2 − 1) 1 Determine the real numbers a , b and c such that, for any x> 1 : g ( x ) = a x + b x + 1 + c x − 1 2 Find a primitive G of g on the interval 1 ; + ∞ . https://chingmath.fr chapExoCorrec/5214 sacados/5214 chapExoCorrec/5211 sacados/5211 chapExoCorrec/5221 sacados/5221 chapExoCorrec/3931 sacados/3931 chapExoCorrec/5210 sacados/5210 sacados/10393 chapExoCorrec/6006 sacados/6006 chapExoCorrec/6005 sacados/6005 chapExoCorrec/5212 sacados/5212 chapExoCorrec/5222 sacados/5222
E.6004 Consider the function f defined on R by the relation: f ( x ) = 4 · e 2 x +2 − e x +1 e 2 x +2 − 1 1 Show that the function f admits for expression : f ( x ) = 3 · e 2 x +2 e 2 x +2 − 1 + e x +1 e x +1 + 1 (Hint : remember to factor e 2 x +2 − 1 ) 2 Deduce the expression of a primitive of the function f . 7. Determining a primitive with initial condition E.5223 For each question, determine the primi-tive of the function that satisfies the given condition : a f ( x ) = x 2 · ( x 3 + 2) 4 ; F ( − 1) = 1 b g ( x ) = x 3 x 2 − 2 ; G (3) = ln 5 c h ( x ) = x x ; H (4) = 3 d j ( x ) = ( x + 1)e x 2 +2 x ; J (1) = e 3 E.3950 For each question, determine the primi-tive of the function verifying the proposed condition : a f ( x ) = x 2 − 2 · x + 4 x ; F (1) = 2 b g ( x ) = x · e x 2 ; G (1) = 3 · e c h ( x ) = 5 (4 · x − 3) 2 ; H (1) = 1 d j ( x ) = 2 · x − 3 x 2 − 2 x + 1 ; J (0) = − 2 E.6917 Consider the two functions f and g de-fined on R by: f ( x )=ln x 2 +1 ; g ( x )=ln 2 · x 2 +2 1 Determine the image of 0 by each of these two functions. 2 Establish that these two functions are primitives of the same function, which we will specify. 8. Differential equations: examples of solutions E.3647 Consider the differential equation : ( E ): y + y =e − x Show that the function u defined on the set of real numbers R by u ( x )= x · e − x is a solution of the differential equation. E.3663 1 Determine the expression of the function f verifying the following differential equation : f = f f (0) = 2 Justify your answer. 2 Determine the expression of the function g verifying the following differential equation : g = 2 · g g (0) = 1 Justify your answer. 3 Determine the expression of the function h verifying the following differential equation : h = 2 · h h (0) = 2 Justify your answer. E.3664 1 Consider the differential equation : ( E ) : y + y = e − x Show that the function u defined on the set of real num-bers R by: u ( x ) = x · e − x is a solution of the differential equation ( E ) . 2 Let f be the function defined on R by: f ( x ) = 9 2 · e − 2 x − 3 · e − 3 x Show that the function f verifies the differential equa-tion : y + 2 y = 3 · e − 3 x https://chingmath.fr chapExoCorrec/6004 sacados/6004 chapExoCorrec/5223 sacados/5223 chapExoCorrec/3950 sacados/3950 chapExoCorrec/6917 sacados/6917 sacados/3647 chapExoCorrec/3663 sacados/3663 chapExoCorrec/3664 sacados/3664
-2-1I-12JOCfCgChCk E.3673 We seek to determine the set of functions f , defined and derivable on the interval 0 ; + ∞ verifying the condition ( E ) : for any strictly positive real number x : x · f ( x ) − f ( x ) = x 2 · e 2 x 1 Show that if a function f , defined and derivable on the in-terval 0 ; + ∞ , verifies the condition ( E ) , then the func-tion g defined on the interval 0 ; + ∞ by: g ( x ) = f ( x ) x verifies : ( E ) : pour any real number x strictly positive : g ( x ) = e 2 x 2 Conjecture the expression of the functions g verifying: g ( x ) = e 2 x 3 What is the function defined and derivable on the inter-val 0 ; + ∞ that verifies the condition ( E ) and cancels at 1 2 ? E.3690 For any real k positive or zero, con- sider the function f k defined on R by: f k ( x ) = x + 1 − k · e x 1 + k · e x 1 Justify that, for any real k positive or zero, the function f k is a solution of the differential equation : ( E ) : 2 · y = ( y − x ) 2 + 1 . 2 Deduce the direction of variations of f k on R . E.8474 Consider the differential equation : y + 3 · y x = 1 and the function f defined on R ∗ by: f ( x )= 1 x 3 + x 4 Show that the function f is a solution of this differential equa-tion. E.8475 Consider the differential equation : 2 · y + x · y =0 Show that the function f defined on R below is a solution of this differential equation : f ( x ) = e − x 2 4 9. Differential equations: y = ay E.3679 Solve differential equations on R : a y = − 3 y b y − y = 0 c 5 y − 2 y = 0 d y = − 3 y E.3680 For each question, determine the value of a ∈ R so that the function f is a solution of the differential equation : y = a · y a f ( x ) = − 3 · e 4 x b f ( x ) = 4 · e 0.2 x E.3681 Determine the solutions of the following differential equations : a y − 3 y = 0 ; f (0) = 2 b 2 y + 3 y = 0 ; f (0) = − 1 c 3 y − 2 y = 0 ; f 3 2 = 2 d y − 3 y = 0 ; f (6) = e 3 E.3683 Four representative curves of functions verifying the differential equation : are plotted in the frame below y = a · y for a ∈ R By observing the tangents to these curves at the point of ab-scissa 0 , determine the differential equation verified by each of these functions. https://chingmath.fr chapExoCorrec/3673 sacados/3673 chapExoCorrec/3690 sacados/3690 chapExoCorrec/8474 sacados/8474 chapExoCorrec/8475 sacados/8475 chapExoCorrec/3679 sacados/3679 chapExoCorrec/3680 sacados/3680 chapExoCorrec/3681 sacados/3681 chapExoCorrec/3683 sacados/3683 -2-1I-12JOCfCgChCk
-2-1I-12JOCfCgChCk E.3682 Four representative curves of functions verifying the differential equation : are plotted in the reference frame below y = y Determine the initial conditions defining each of its functions. 10. Differential equations: y = ay + b E.3692 Solve the following differential equa-tions : a y + y = 2 b y − 3 · y = − 3 c 6 · y = 3 · y + 2 d 5 · y = 3 2 · y + 1 3 E.3693 Solve the following differential equations : a 4 · y − y = 4 ; y (1) = e b 15 · y + 24 · y = 12 ; y 5 4 = 2 c − 3 2 · y + 1 4 · y = − 1 ; y (3) = 6 + 2 · e 11. Differential equations: y = ay + f E.3685 Consider the function f defined on the interval 0 ; + ∞ by: f ( x ) = ln e 2 x − 1 e x Verify that f is a solution of the differential equation : y + y = e x e x − 1 − e x e x + 1 E.3686 Let f be the function defined on R by: f ( x ) = e − 2 x · ln 1 + 2 · e x Consider the differential equation : ( E ) : y + 2 · y = 2 · e − x 1 + 2 · e x 1 Verify that the function f is a solution of ( E ) . 2 Show that a function ’ is a solution of ( E ) if, and only if, ’ − f is a solution of the differential equation : ( E ) : y + 2 y = 0 . 3 Solve ( E ) and deduce the solutions of ( E ) . E.4304 Consider the two differential equa-tions : ( E ) : y + y = e − x ( E ) : y + y = 0 1 Show that the function u defined on the set of real num-bers R by u ( x )= x · e − x is a solution of the differential equation ( E ) . 2 Solve the differential equation ( E ) . 3 Let v be a function defined and derivable on R . Show that the function v is a solution of the differential equa-tion ( E ) if, and only if, the function v − u is a solution of the differential equation ( E ) . 4 Deduce all solutions of the differential equation ( E ) . 5 Determine the unique solution g of the differential equa-tion ( E ) such that g (0) = 2 . https://chingmath.fr chapExoCorrec/3682 sacados/3682 -2-1I-12JOCfCgChCk chapExoCorrec/3692 sacados/3692 chapExoCorrec/3693 sacados/3693 Extrait de repere - Hachette chapExoCorrec/3685 sacados/3685 chapExoCorrec/3686 sacados/3686 Antilles-Guyane Septembre 2000 sacados/4304
E.3655 Let f be the function defined on R by: f ( x ) = 9 2 · e − 2 x − 3 · e − 3 x Let the differential equation be : ( E ): y +2 y =3e − 3 x . 1 Solve the differential equation : ( E ): y +2 y =0 . 2 Deduce that the function h defined on R by: h ( x ) = 9 2 · e − 2 x is solution of ( E ) . 3 Verify that the function g defined on R by: g ( x ) = − 3e − 3 x is solution of equation ( E ) . 4 Noting that f = g + h , show that f is a solution of ( E ) . E.3678 The function f is defined on the interval 0 ; + ∞ by: f ( x ) = 20 x + 10 · e − 1 2 x We note y ( t ) the value, in degrees Celsius, of the temperature of a chemical reaction at time t , t being expressed in hours. The initial value, at time t =0 , is y (0)=10 . We admit that the function which, to any real t belonging to the interval 0 ; + ∞ associates y ( t ) , is solution of the differ-ential equation ( E ) : ( E ) : y + 1 2 y = 20 · e − 1 2 t 1 Verify that the function f is a solution of the differential equation ( E ) on the interval 0 ; + ∞ . 2 We propose to show that this function f is the unique solution of the differential equation ( E ) , defined on the interval 0 ; + ∞ , which takes the value 10 at time 0 . a Let g be any solution of the differential equation ( E ) , defined on 0 ; + ∞ verifying g (0)=10 . Show that the function g − f is a solution, on the interval 0 ; + ∞ of the differential equation : ( E ) : y + 1 2 · y = 0 b Solve the differential equation ( E ) . c Conclude. E.3684 Part A - Solving a differential equation Consider the differential equation : y − 2 y = e 2 x ; ( E ) 1 Demonstrate that the function u defined on R by: u ( x ) = x · e 2 x is a solution of ( E ) . 2 Solve the differential equation : y − 2 · y = 0 ( E 0 ) 3 Show that a function v defined on R is a solution of ( E ) if and only if v − u is a solution of ( E 0 ) . 4 Deduce all solutions of the equation ( E ) . 5 Determine the function, solution of ( E ) , which takes the value 1 in 0 . Part B - Study of a function The plane is referred to the orthonormal reference frame O ; −→ i ; −→ j . Let the function f be defined on R by: f ( x ) = x + 1 · e 2 x Note C the representative curve of f in the reference frame O ; −→ i ; −→ j . 1 Study the limit of f in + ∞ then the limit of f in −∞ . 2 Let x be a real number. Calculate f ( x ) . Study the variations of f then draw up its table of vari-ations. Specify the sign of f ( x ) for any real x . Part C - Solving an equation 1 Show that the equation f ( x )=2 admits a unique solution x 0 in the interval 0.2 ; 0.3 . 2 Copy and complete the following table : x 0.05 0.1 0.15 0.2 0.25 0.3 f ( x ) Values of f ( x ) will be rounded to within 10 − 2 by default. 3 On the graph paper below, où units are 10 cm in abscissa and 5 cm in ordinate, draw the arc of the curve C for x belonging to 0 ; 0.3 . Show x 0 on the graph. E.3687 We seek to solve the differential equation : (1) : y − 2 y = x · e x 1 Solve the differential equation : (2) : y − 2 y = 0 , où y denotes a function derivable on R . 2 Let a and b be two real numbers and let u be the function defined on R by: u ( x ) = a · x + b · e x a Determine a and b so that u is a solution to the equa-tion (1) . b Show that v is a solution of equation (2) if, and only if, u + v is a solution of (1) . c Deduce the set of solutions of (1) . 3 Determine the solution of equation (1) that cancels at 0 . E.3966 Consider the differential equation : ( E ): y =2 · y +cos x 1 Determine two real numbers a and b such that the func-tion f 0 defined on R by: f 0 ( x )= a · cos x + b · sin x or a solution f 0 of ( E ) . 2 Solve the differential equation : ( E 0 ) : y =2 · y . 3 Demonstrate that f is a solution of ( E ) if, and only if, f − f 0 is a solution of ( E 0 ) . 4 Deduce the solutions of ( E ) . 5 Determine the solution k of ( E ) verifying k ı 2 =0 E.3689 Consider a differential equation : ( E ) : y − 3 y = − 3 e 1 + e − 3 x 2 We give a function ’ derivable on R and the function f de-fined on R by: f ( x )=e − 3 x · ’ ( x ) . 1 Show that f is derivable on R and for any real x , express ’ ( x ) − 3 · ’ ( x ) as a function of f ( x ) . 2 Determine f so that ’ is a solution of ( E ) on R and verifies : ’ (0)= e 2 . https://chingmath.fr sacados/3655 Extrait Antilles Guyane Juin 2008 sacados/3678 Extrait France Septembre 2005 chapExoCorrec/3684 sacados/3684 chapExoCorrec/3687 sacados/3687 Antilles-uyane Juin 2001G chapExoCorrec/3966 sacados/3966 Extrait Amerique du Sud Novembre 2007 sacados/3689 Extrait France Septembre 2003
12. Other differential equations E.3676 Two different ways are sought to model the evolution of the number, expressed in millions, of French households owning a flat-screen TV, as a function of the year. Let g ( x ) be the number, expressed in millions, of such house-holds in year x . We pose x =0 in 2005 , g (0)=1 and g is a solution, which does not cancel on 0 ; + ∞ of the differential equation : ( E ) : y = 1 20 · y · (10 − y ) 1 Consider a function y that does not cancel at 0 ; + ∞ and set z = 1 y a Show that y is a solution of ( E ) if, and only if, z is a solution of the differential equation : ( E 1 ) : z = − 1 2 · z + 1 20 b Solve the equation ( E 1 ) and deduce the solutions of the equation ( E ) . 2 Show that g is defined on 0 ; + ∞ by: g ( x ) = 10 9e − 1 2 x + 1 3 Study the variations of g on 0 ; + ∞ . 4 Calculate the limit of g in + ∞ and interpret the result. 5 In what year will the number of households with such equipment exceed 5 million? E.3691 In reality, in an observed area of a given region, a predator prevents such growth by killing a certain quantity of rodents. We denote u ( t ) the number of rodents alive at time t (expressed in years) in this region, and we admit that the function u , thus defined, satisfies the conditions : ( E 2 ) u ( x ) = u ( t ) 4 − u ( t ) 2 12 pour t ∈ R ∗ + u (0) = 1 où u denotes the function derived from the function u . 1 It is assumed that, for any positive real t , we have u ( t ) > 0 . Consider, on the interval 0 ; + ∞ , the function h defined by h = 1 u . Show that the function u satisfies the condi-tions ( E 2 ) if, and only if, the function h satisfies the conditions : ( E 3 ) h ( t ) = − 1 4 · h ( t ) + 1 12 pour t ∈ R ∗ + h (0) = 1 où h denotes the function derived from the function h . 2 Give the solutions of the differential equation : y = − 1 4 · y + 1 12 and deduce the expression of the function h , then that of the function u . 3 In this model, how does the size of the study population behave when t tends to + ∞ ? E.3773 We propose to determine all func-tions f defined and derivable on the interval 0 ; + ∞ verify-ing the differential equation : ( E ) : x · f ( x ) − (2 x +1) · f ( x ) = 8 · x 2 1 a Show that if f is solution of ( E ) then the function g defined on the interval 0 ; + ∞ by: g ( x ) = f ( x ) x is the solution of the differential equation : ( E ) : y = 2 · y + 8 b Show that if h is a solution of ( E ) then the function f defined by f ( x )= x · h ( x ) is a solution of ( E ) . 2 Solve ( E ) and deduce all solutions of ( E ) . E.3646 1 Let f be a function defined and derivable on the interval 0 ; + ∞ verifying for any strictly positive real number x : xf ( x ) − f ( x ) = x 2 · e 2 x Let g be the function defined on 0 ; + ∞ by: g ( x )= f ( x ) x Show that for any strictly positive real number x , we have : g ( x ) = e 2 x 2 Consider the function h defined on the interval 0 ; + ∞ by: h ( x ) = 1 2 x e 2 x − e 2 x Determine, according to the values of the positive real number x , the sign of h ( x ) . E.3688 We call ( E ) the differential equa-tion : y  − y = 0 , où y is a numerical function defined and twice derivable on the set R of real numbers. 1 Determine the reals r such that the function h , defined by h ( x )=e r · x , is a solution of ( E ) . 2 Verify that the functions ’ defined by ’ ( x )= ¸ · e x + ˛ · e − x , où ¸ and ˛ are two real numbers, are solutions of ( E ) . We admit that we thus obtain all solutions of ( E ) . 3 Determine the particular solution of ( E ) whose represen-tative curve passes through the point with coordinates ln 2 ; 3 4 and admits at this point a tangent whose slope is 5 4 . 13. Course - old program: Differential equations https://chingmath.fr chapExoCorrec/3676 sacados/3676 chapExoCorrec/3691 sacados/3691 Extrait France Juin 2005 chapExoCorrec/3773 sacados/3773 Metropole et La reunion Septembre 2008 sacados/3646 sacados/3688
E.3283 Prerequisites: The solutions of the differential equation y = − –y are the functions : x ↦−→ C e − λx où C is a real constant. 1 Demonstrate the existence and uniqueness of the solu-tion z of the differential equation ( E λ ): z = − ( –z +1) such that z (0) = 1 2 Give the expression of this function, which will be de- noted z 0 . E.3284 Consider the differential equation ( E ) : y = 1 16 · y 1 It is known that the function x ↦−→ e x 16 is a solution of the differential equation ( E ) . Show then that the set of solutions of the equation ( E ) is the set of functions, de-fined on R , of the form x ↦−→ K · e x 16 , où K is any real number. 2 Show that there is a unique solution of the differential equation ( E ) taking the value − 4 in 0 . 14. Unclassified financial years E.10394 Consider the function f defined on R ∗ + by: f ( x ) = 2 · x · ln( x ) The primite F of the function f such that F (1) = − 1 2 admits as expression : a F ( x ) = x 2 · ln( x ) − 1 2 · x b F ( x ) = x · ln( x ) − 1 2 · x 2 c F ( x ) = x · ln( x ) − 1 2 · x d F ( x ) = x 2 · ln( x ) − 1 2 · x 2 E.10395 Consider the function f defined on R ∗ + by: f ( x ) = x 2 − 2 · e x The primite F of the function f such that F (1) = − 1 2 admits as expression : a F ( x ) = x 2 − 2 · e x b F ( x ) = x 2 − 2 · x · e x c F ( x ) = x 3 − 2 · e x d F ( x ) = x 3 − 2 · x · e x https://chingmath.fr chapExoCorrec/3283 sacados/3283 Extrait de France Septembre 2006 chapExoCorrec/3284 sacados/3284 sacados/10394 sacados/10395