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 2021-02-12 Answered
On solution of the differetial equation y+y=0 is y=ex. Use Reduction of Order to find a second linearly independent solution.
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bahaistag
Answered 2021-02-13 Author has 101 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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