## Ordinary differential equations e**xercises **

**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 exercises** 2: **

Let the differential equation be defined on ℝ by phrase:

**y'+y=0 ..........(2)**on ℝ Where y is a derivable function on ℝ.

*f*Where the function

*f*is the solution of the differential equation (1) and Check that

*f(0)=1.*

## Differential equations exercises** 3: **

Let us be the following two differential equations:

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

**(****E′):y′−2y=1−exsin(x)**

**h**

**(x)=3e2x−12**

**k**

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

**Exercises of differential equations 4: **

Let the following differential equation :

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

**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

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