Chapter D2.1, Problem 70E

Reduction of order Suppose you are solving a second-order linear homogeneous differential equation and you have found one solution. A method called reduction of order allows you to find the second (linearly independent) solution (up to evaluating integrals). Consider the differential equation $y^{″}-\frac{1}{t}{y}^{\prime }+\frac{1}{t^2}y=0$, for t > 0.

a.    Verify that y₁ = t is a solution. Assume the second homogeneous solution is y₂ and it has the form $y_2\left(t\right)=v\left(t\right)y_1\left(t\right)=v\left(t\right)t$, where v is a function to be determined.

b.    Substitute y₂ into the differential equation and simplify the resulting equation to show that v satisfies the equation $v^{″}=-\frac{v^{\prime }}{t}$.

c.    Note that this equation is first order in v′; so let w = v′ to obtain the first-order equation $w^{\prime }=-\frac{w}{t}$.

d.    Solve this separable equation and show that $w=\frac{c_1}{t}$.

e.    Now solve the equation $v^{\prime }=w=\frac{c_1}{t}$ to find v.

f.    Finally, recall that y₂(t) = v(t)t and conclude that the second solution is y₂(t) = c₁ t ln t.