On solution of the differetial equation y''+y'=0 is y=e^{-x}. Use Reduction of Order to find a second linearly independent solution.

Tahmid Knox

Tahmid Knox

Answered question

2021-02-12

On solution of the differetial equation y+y=0 is y=ex. Use Reduction of Order to find a second linearly independent solution.

Answer & Explanation

bahaistag

bahaistag

Skilled2021-02-13Added 100 answers

In such problems ie second order linear differential equations when we are given one solution, y1 we assume the second solution to be of the form, y2=vy1 and substitute y2 in the given ode and reduce order of the differential equation by using the fact that y1 is a solution.
Compute:y2,y2
y2=vy"1+vy1
=vex+vex
=ex(v+v)
y2=ex(v+v)+ex(v+v)
=ex(vvv+v)
Substitute y2,y2 in given differential equation
ex(v2v+v)+ex(v+v)=0
ex(v+v)=0
v+v=0
Let u=v we get a first order differential equation
u+u=0
u=u
u=ex
Substitute u=v in above equation and solve for v
v=ex
Integrating we get
v=ex
Get second linearly independent solution by substituting v in expression for y2
y2=vex
=cexex
=c
Hence second linearly independent solution is c

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