# Differential Equations exercises

## Exercises of differential equations 1:

We consider the differential equation

(E):y+y=ex

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

F(x)=xex

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:

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   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   is the solution of the differential equation (1) and Check that f(0)=1.
Calculate in terms of n,the total Sn=W0+W1+..........+Wn-1

## Differential equations exercises 3:

Let us be the following two differential equations:

(E):    y'-2y-1=0

(E):y2y=1exsin(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)=3e2x12

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)=ce23x

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 :
xce5x+7

### - 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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