Differential Equations | Part IA, 2008

The linear second-order differential equation

d2ydx2+p(x)dydx+q(x)y=0\frac{d^{2} y}{d x^{2}}+p(x) \frac{d y}{d x}+q(x) y=0

has linearly independent solutions y1(x)y_{1}(x) and y2(x)y_{2}(x). Define the Wronskian WW of y1(x)y_{1}(x) and y2(x)y_{2}(x).

Suppose that y1(x)y_{1}(x) is known. Use the Wronskian to write down a first-order differential equation for y2(x)y_{2}(x). Hence express y2(x)y_{2}(x) in terms of y1(x)y_{1}(x) and WW.

Show further that WW satisfies the differential equation

dWdx+p(x)W=0\frac{d W}{d x}+p(x) W=0

Verify that y1(x)=x22x+1y_{1}(x)=x^{2}-2 x+1 is a solution of

(x1)2d2ydx2+(x1)dydx4y=0.(x-1)^{2} \frac{d^{2} y}{d x^{2}}+(x-1) \frac{d y}{d x}-4 y=0 .

Compute the Wronskian and hence determine a second, linearly independent, solution of ()(*).

Typos? Please submit corrections to this page on GitHub.