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.

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

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

$y_2=x\ln x\implies{y_1}'=\ln x+1\implies{y_1}''=\dfrac1x$

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

Both solutions satisfy the ODE.

b. The Wronskian determinant is

$\begin{vmatrix}x&x\ln x\\1&\ln x+1\end{vmatrix}=x(\ln x+1)-x\ln x=x\neq0$

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,

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