Differential Equations – Changing 2nd Order Homogeneous Differential Equation to Constant Coefficients

ordinary differential equations

Given a general 2nd degree homogeneous differential equation $y'' +p(x)y' +q(x)y=0$.
Change the independent variable from $x$ to $z=z(x)$. Show that the above given homogeneous 2nd order differential equation can be transformed into an equation with constant coefficients if and only if $(q' + 2pq)/q^{3/2}$ is constant.

Best Answer

When we change variables from $x$ to $z(x)$ what we're doing is effectively looking for a function $z$ that shifts $y$ and its derivatives so that $p(x)$ and $q(x)$ are fixed relative to them, allowing us to consider $p$ and $q$ as constants in this new variable.

By the chain rule, \begin{equation} \frac{\mathrm d}{\mathrm{d}x} y(z(x)) = \frac{\mathrm d}{\mathrm{d}z} y(z) \frac{\mathrm{d}z}{\mathrm{d}x} \end{equation} and \begin{equation} \frac{\mathrm d}{\mathrm{d}x} (\frac{\mathrm d}{\mathrm{d}z} y(z) \frac{\mathrm{d}z}{\mathrm{d}x}) = \frac{\mathrm{d}^2}{\mathrm{d}z^2} y(z) (\frac{\mathrm{d}z}{\mathrm{d}x})^2 +\frac{\mathrm d}{\mathrm{d}z} y(z) \frac{\mathrm{d}^2 z}{\mathrm{d}x^2} \end{equation}.

Substituting into the original equation and rearranging, we get

\begin{equation} y''(z) +y'(z)(\frac{z''(x) +p(x)z'(x)}{(z'(x))^2}) + \frac{q(x)}{(z'(x))^2} y(z) = 0 \end{equation}

And so for the new coefficients to be constant, we require $\frac{z''(x) +p(x)z'(x)}{(z'(x))^2}= A_1$ and $\frac{q(x)}{(z'(x))^2} = \frac{1}{A_2}$ where $A_1,A_2$ are constants.

the second of these is easier to solve, giving us $z'(x) = \sqrt{A_2q(x)}$, which we can differentiate to give \begin{equation}z''(x) = \frac{1}{2}\frac{A_2}{\sqrt{A_2 q(x)}} q'(x) \end{equation} We can then plug this into the equation for the other coefficient and solve for $z$, \begin{equation} \frac{\frac{1}{2}\frac{A_2}{\sqrt{A_2 q(x)}} q'(x) +p(x)\sqrt{A_2 q(x)}}{A_2 q(x)} = A_1 \end{equation} Multiplying the top and bottom of the LHS by $\sqrt{A_2 q(x)}$ and rearranging gives us \begin{equation} \frac{q'(x) + 2p(x)q(x)}{q(x)^\frac{3}{2}} = 2\sqrt{A_2}A_1 \end{equation} and so the desired expression is clearly constant.

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