Home » Blog » Prove a substitution converts a given second order equation to a first order equation

Prove a substitution converts a given second order equation to a first order equation

  1. Consider the second-order differential equation

        \[ y'' + P(x) y' + Q(x) y = 0 \]

    and let u be a solution to the equation. Show that the substitution y = uv converts the equation

        \[ y''  + P(x)y' + Q(x)y = R(x) \]

    into a first-order liner equation for v'.

  2. By inspection, find a nonzero solution of the second order differential equation

        \[ y'' - 4y' + x^2 (y' - 4y) = 0 \]

    and use part (a) to find a solution of the differential equation

        \[ y'' - 4y' + x^2(y' - 4y) = 2xe^{-\frac{x^3}{3}} \]

    with

        \[ y = 0 \quad \text{and} \quad y' = 4 \quad \text{when } x = 0. \]


Incomplete.

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):