1.91 problem 89

Internal problem ID [7783]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 89
Date solved : Tuesday, October 22, 2024 at 02:29:51 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

Solve

\begin{align*} y^{\prime \prime }-y^{\prime } y&=2 x \end{align*}

1.91.1 Maple step by step solution

1.91.2 Maple trace

Methods for second order ODEs:
 

1.91.3 Maple dsolve solution

Solving time : 0.029 (sec)
Leaf size : 147

dsolve(diff(diff(y(x),x),x)-diff(y(x),x)*y(x) = 2*x, 
       y(x),singsol=all)
 

\[ y = \frac {-\operatorname {WhittakerM}\left (\frac {i \sqrt {2}\, c_1}{8}+1, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right ) \left (6+i \sqrt {2}\, c_1 \right )+8 c_2 \operatorname {WhittakerW}\left (\frac {i \sqrt {2}\, c_1}{8}+1, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )+2 \left (1-i \left (x^{2}-\frac {c_1}{2}\right ) \sqrt {2}\right ) \left (c_2 \operatorname {WhittakerW}\left (\frac {i \sqrt {2}\, c_1}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )+\operatorname {WhittakerM}\left (\frac {i \sqrt {2}\, c_1}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )\right )}{2 x \left (c_2 \operatorname {WhittakerW}\left (\frac {i \sqrt {2}\, c_1}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )+\operatorname {WhittakerM}\left (\frac {i \sqrt {2}\, c_1}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )\right )} \]

1.91.4 Mathematica DSolve solution

Solving time : 45.186 (sec)
Leaf size : 318

DSolve[{D[y[x],{x,2}]+D[y[x],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*}