Internal problem ID [7135]
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]]
\[ \boxed {y^{\prime \prime }-y y^{\prime }=2 x} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 161
dsolve(diff(y(x),x$2)-diff(y(x),x)*y(x)=2*x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-\operatorname {WhittakerM}\left (\frac {i c_{1} \sqrt {2}}{8}+1, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right ) \left (6+i c_{1} \sqrt {2}\right )+8 c_{2} \operatorname {WhittakerW}\left (\frac {i c_{1} \sqrt {2}}{8}+1, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )+2 \left (1-i \left (x^{2}-\frac {c_{1}}{2}\right ) \sqrt {2}\right ) \left (c_{2} \operatorname {WhittakerW}\left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )+\operatorname {WhittakerM}\left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )\right )}{2 x \left (c_{2} \operatorname {WhittakerW}\left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )+\operatorname {WhittakerM}\left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 42.411 (sec). Leaf size: 318
DSolve[y''[x]+y'[x]*y[x]==2*x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt [4]{2} \left (\sqrt [4]{2} x \operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (-\sqrt {2} c_1-2\right ),i \sqrt [4]{2} x\right )+2 i \operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (2-\sqrt {2} c_1\right ),i \sqrt [4]{2} x\right )+c_2 \left (2 \operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (\sqrt {2} c_1+2\right ),\sqrt [4]{2} x\right )-\sqrt [4]{2} x \operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (\sqrt {2} c_1-2\right ),\sqrt [4]{2} x\right )\right )\right )}{\operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (-\sqrt {2} c_1-2\right ),i \sqrt [4]{2} x\right )+c_2 \operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (\sqrt {2} c_1-2\right ),\sqrt [4]{2} x\right )} \\ y(x)\to \sqrt {2} x-\frac {2 \sqrt [4]{2} \operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (\sqrt {2} c_1+2\right ),\sqrt [4]{2} x\right )}{\operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (\sqrt {2} c_1-2\right ),\sqrt [4]{2} x\right )} \\ y(x)\to \sqrt {2} x-\frac {2 \sqrt [4]{2} \operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (\sqrt {2} c_1+2\right ),\sqrt [4]{2} x\right )}{\operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (\sqrt {2} c_1-2\right ),\sqrt [4]{2} x\right )} \\ \end{align*}