Ordinary differential equations 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 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 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 exercises 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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