Question

Verify that y1 = x and y2 = x ln x are solutions to x 2y ′′ − xy′ + y = 0. b) Use the Wronskian to show that y1 and y2 are linearly independent. c) Find the particular solution to the differential equation with initial conditions y(1) = 7, y′ (1) = 2

Answer 1

a. Substitute the given solutions and their derivatives into the ODE.

`y_1=x\implies {y_1}'=1\implies{y_1}''=0`

`x^2y''-xy'+y=-x+x=0`

`y_2=x\ln x\implies{y_1}'=\ln x+1\implies{y_1}''=\frac1x`

`x^2y''-xy'+y=x-x(\ln x+1)+x\ln x=0`

Both solutions satisfy the ODE.

b. The Wronskian determinant is

`\begin{vmatrix}x&x\ln x\\1&\ln x+1\end{vmatrix}=x(\ln x+1)-x\ln x=x\\eq0`

so the solutions are indeed independent.

c. The ODE has general solution
`y(t)=C_1x+C_2x\ln x`. Then with the given initial conditions, the constants satisfy

`y(1)=7\implies 7=C_1`

`y'(1)=2\implies2=C_1+C_2\implies C_2=-5`

So the ODE has the particular solution,

`\boxed{y(t)=7x-5x\ln x}`