2.1033   ODE No. 1033

1. Problem in Latex
2. Mathematica input
3. Maple input

\[ a e^{-2 x} y(x)+y''(x)+y'(x)=0 \] ✓ Mathematica : cpu = 0.0129288 (sec), leaf count = 37

\[\left \{\left \{y(x)\to c_1 \cos \left (\sqrt {a} e^{-x}\right )-c_2 \sin \left (\sqrt {a} e^{-x}\right )\right \}\right \}\] ✓ Maple : cpu = 0.019 (sec), leaf count = 27

\[\left \{y \left (x \right ) = c_{1} \sin \left (\sqrt {a}\, {\mathrm e}^{-x}\right )+c_{2} \cos \left (\sqrt {a}\, {\mathrm e}^{-x}\right )\right \}\]

Hand solution

\[ y^{\prime \prime }+y^{\prime }+ae^{-2x}y=0 \]

Let \(y\left ( x\right ) =\eta \left ( \xi \right ) \) where \(\xi =e^{-x}\), hence

\begin {align*} \frac {dy}{dx} & =\frac {d\eta }{d\xi }\frac {d\xi }{dx}\\ & =\frac {d\eta }{d\xi }\left ( -e^{-x}\right ) \end {align*}

And

\begin {align*} \frac {d^{2}y}{dx^{2}} & =\frac {d}{dx}\left ( \frac {d\eta }{d\xi }\left ( -e^{-x}\right ) \right ) \\ & =\frac {d^{2}\eta }{d\xi ^{2}}\frac {d\xi }{dx}\left ( -e^{-x}\right ) +\frac {d\eta }{d\xi }\left ( e^{-x}\right ) \\ & =\frac {d^{2}\eta }{d\xi ^{2}}\left ( -e^{-x}\right ) \left ( -e^{-x}\right ) +\frac {d\eta }{d\xi }\left ( e^{-x}\right ) \\ & =\frac {d^{2}\eta }{d\xi ^{2}}\left ( e^{-2x}\right ) +\frac {d\eta }{d\xi }\left ( e^{-x}\right ) \end {align*}

Hence the original ODE becomes

\begin {align*} \frac {d^{2}\eta }{d\xi ^{2}}\left ( e^{-2x}\right ) +\frac {d\eta }{d\xi }\left ( e^{-x}\right ) +\frac {d\eta }{d\xi }\left ( -e^{-x}\right ) +ae^{-2x}\eta \left ( \xi \right ) & =0\\ \eta ^{\prime \prime }+a\eta & =0 \end {align*}

This is standard second order with constant coefficients. The solution is

\[ \eta =c_{1}\cos \left ( \sqrt {a}\xi \right ) +c_{2}\sin \left ( \sqrt {a}\xi \right ) \]

Substituting back

\[ y\left ( x\right ) =c_{1}\cos \left ( \sqrt {a}e^{-x}\right ) +c_{2}\sin \left ( \sqrt {a}e^{-x}\right ) \]

Verification

restart; 
ode:=diff(diff(y(x),x),x)+diff(y(x),x)+a*exp(-2*x)*y(x)=0; 
ys:=_C1*cos(sqrt(a)*exp(-x))+_C2*sin(sqrt(a)*exp(-x)); 
odetest(y(x)=ys,ode); 
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