Answer to Question #326598 in Differential Equations for Allen

Question

Determine the unique solution of the initial value problem. 

1. y′′ −2y′ +y=0,y(0)=7,y′(0)=4.

Answer 1

Solution

For given homogeneous equation the characteristic equation is

λ² - 2λ + 1=0 => (λ-1)² =>  λ_1,2 = 1

So the solution of homogeneous equation is y(x) = C₁e^x+C₂xe^x, where C₁, C₂ are arbitrary constants.

From initial conditions y(0)=7,y′(0)=4 we’ll get

C₁ = 7, C₁ + C₂ = 4 => C₁ = 7, C₂ = -3

So the unique solution is y(x) = 7e^x – 3xe^x