Integration exercises with answers

Ordinary differential equation exercises 

Exercises of differential equations 1: 

We consider the differential equation   

                                                    (E):y′+y=e−x

1-Prove that the function F defined is on ℝ by

                                                    F(x)=xe−x

Represent a solution to the differential equation

2-We consider the differential equation  (E');y'+y=0

    Solve the differential equation (E') on ℝ 

3-Let the function G defined and derivable on ℝ. 

a) Prove that the function G is a solution to the differential equation (E) If and only if function G-F is a solution to the differential equation (E').

b) conclude all differential equation solutions.

4-Determine the special solution K of the differential equation (E) where K(0)=2

Differential equations exercise 2: 

Let the differential equation be defined on ℝ by phrase:

1- Solve the differential equation y'+y=0  ..........(2) on ℝ Where y is a derivable function on ℝ.

2-

 a)Prove that the solution to the differential equation (1) is the function U defined on ℝ  by:

b) Prove that if the function V is a solution to the differential equation (2) then U+V  is the solution of the differential equation (1)

c) conclude the group solutions of differential equation (1).

3- Determine the function f  Where the function  f  is the solution of the differential equation (1) and Check that f(0)=1.

4- We put for every natural number n: 

                                                                    

Calculate in terms of n, the total Sn=W0+W1+..........+Wn-1

Differential equations exercise 3: 

Let us be the following two differential equations:

                                                         (E):    y'-2y-1=0                           

                                         (E′):y′−2y=1−exsin(x)

                                                      

Answer true or false with the reasoning:

1- The differential equation (E) accepts a polynomial function as a solution.

2- Let it be a positive function g defined on ℝ, If g is a solution to the differential equation  (E) then g  is increasing on ℝ. 

3- The function h defined on ℝ is a solution to the differential equation (E), where:                                                                                                                                                                      

                                                            h(x)=3e2x−12

4-   The function k defined on ℝ is a solution to the differential equation (E'), where: : 

                                             k(x)=ex2[cos(x)+sin(x)]

Exercises of differential equations 4: 

Answer true or false with the reasoning:

Let the following differential equation :

                                                         (E):    3y'+2y=0                  

1- Solutions to the differential equation (E) are the functions that are written in the figure:

                                                      f(x)=ce−23x

2-  If it was f(-3)=√e  Then the differential equation (E) accepts a single-scaled solution as follows.

3- The differential equation Solutions y'+5y=35 is the functions :

                                                            x↦ce−5x+7

objectives of the exercises: 

- Solving differential equations

- Find the special solution to the differential equation

- Determine the functions that represent all solutions of the differential equation

- Ensure that a function is a solution to the differential equation

- Ordinary differential equations practice problems

These exercises help you understand differential equations and prepare you to apply them in various fields.

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Integration exercises with answers

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