Internal problem ID [6382]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 89.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-y y^{\prime }-2 x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.115 (sec). Leaf size: 227
dsolve(diff(y(x),x$2)-diff(y(x),x)*y(x)=2*x,y(x), singsol=all)
\[ y \relax (x ) = \frac {4 c_{2} \WhittakerW \left (\frac {i c_{1} \sqrt {2}}{8}+1, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )}{x \left (c_{2} \WhittakerW \left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )+\WhittakerM \left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )\right )}+\frac {\left (-2 i \sqrt {2}\, c_{2} x^{2}+i \sqrt {2}\, c_{1} c_{2}+2 c_{2}\right ) \WhittakerW \left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )+\left (-i c_{1} \sqrt {2}-6\right ) \WhittakerM \left (\frac {i c_{1} \sqrt {2}}{8}+1, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )+\left (-2 i \sqrt {2}\, x^{2}+i c_{1} \sqrt {2}+2\right ) \WhittakerM \left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )}{2 x \left (c_{2} \WhittakerW \left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )+\WhittakerM \left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.035 (sec). Leaf size: 163
DSolve[y''[x]+y'[x]*y[x]==2*x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {2} x-\frac {2 \sqrt [4]{2} \left (\sqrt [4]{2} x D_{\frac {1}{4} \left (-\sqrt {2} c_1-2\right )}\left (i \sqrt [4]{2} x\right )+i D_{\frac {1}{4} \left (2-\sqrt {2} c_1\right )}\left (i \sqrt [4]{2} x\right )+c_2 D_{\frac {1}{4} \left (\sqrt {2} c_1+2\right )}\left (\sqrt [4]{2} x\right )\right )}{D_{\frac {1}{4} \left (-\sqrt {2} c_1-2\right )}\left (i \sqrt [4]{2} x\right )+c_2 D_{\frac {1}{4} \left (\sqrt {2} c_1-2\right )}\left (\sqrt [4]{2} x\right )} \\ \end{align*}