Solve the following Higher Order Non-homogenous DE using

nastupnat0hh

nastupnat0hh

Answered question

2022-03-23

Solve the following Higher Order Non-homogenous DE using Methods of undetermined coefficients
y2y+y=3x+2

Answer & Explanation

Madilyn Shah

Madilyn Shah

Beginner2022-03-24Added 11 answers

Given,
y2y+y=3x+2
A second order linear, non-homogeneous ODE has the form of
ay+by+cy=g(x)
The general solution to a(x)y+b(x)y+c(x)y=g(x) can be written as
y=yh+yp
yh is the solution to the homogeneous ODE a(x)y+b(x)y+c(x)y=0
yp, the particular solution, is any function that satisfies the non-homogeneous equation
Auxiliary equation is:
m22m+1
(m1)2
(m1)(m1)
Which has two real and district solutions m=1,1
the solution of the homogeneous equation is:
yc=c1ex+c2ex
With this particular equation [A], a probable solution is of the form
y=ax+b
Where a , b , c are constants to be determined by substitution
Let us assume the above solution works, in which case be differentiating wrt x we have
y=a
y=0
Substituting into the initial Differential Equation we get:
02a+(ax+b)=3x+2
2a+ax+b=3x+2
Equating coefficients of x0 and x we get
x0: 2a+b=2
23+b=2
b=2+6
b=8
x1: ax=3xa=3
And so we form the Particular solution:
yp=3x+8
General Solution
y(x)=yc+yp
y=c1ex+c2xex+3x+8
Jeffrey Jordon

Jeffrey Jordon

Expert2022-03-31Added 2605 answers

Answer is given below (on video)

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Equations

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?