Answer to Question #88486 in Differential Equations for adigam amos jacob

Question

solve completely the differential equation d^2y/dx^2-a^2y=0

Answer 1

Solution

y^''-a²y=0

Looking for a solution in the form y=e^λx

(e^λx)''-a²e^λx=0

λ²e^λx-a²e^λx=0

λ²-a²=0

λ=a or λ=-a

if a not equal to zero

y₁=c₁e^ax and y₂=c₂e^-ax

y(x)=c₁e^ax+c₂e^-ax

if a=0

y₁=c₁ and y₂=c₂x

y(x)=c₁+c₂x

Answer

y(x)=c₁e^ax+c₂e^-ax , if a not equal to zero

y(x)=c₁+c₂x , if a=0