56.1.91 problem 89
Internal
problem
ID
[8803]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
1.0
Problem
number
:
89
Date
solved
:
Thursday, March 13, 2025 at 06:09:38 PM
CAS
classification
:
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]
\begin{align*} y^{\prime \prime }-y^{\prime } y&=2 x \end{align*}
✓ Maple. Time used: 0.068 (sec). Leaf size: 147
ode:=diff(diff(y(x),x),x)-y(x)*diff(y(x),x) = 2*x;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-\operatorname {WhittakerM}\left (\frac {i c_{1} \sqrt {2}}{8}+1, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right ) \left (6+i c_{1} \sqrt {2}\right )+8 c_{2} \operatorname {WhittakerW}\left (\frac {i c_{1} \sqrt {2}}{8}+1, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )+2 \left (1-i \left (x^{2}-\frac {c_{1}}{2}\right ) \sqrt {2}\right ) \left (c_{2} \operatorname {WhittakerW}\left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )+\operatorname {WhittakerM}\left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )\right )}{2 x \left (c_{2} \operatorname {WhittakerW}\left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )+\operatorname {WhittakerM}\left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )\right )}
\]
✓ Mathematica. Time used: 48.14 (sec). Leaf size: 318
ode=D[y[x],{x,2}]+D[y[x],x]*y[x]==2*x;
ic={};
DSolve[{ode,ic},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*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-2*x - y(x)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -(-2*x + Derivative(y(x), (x, 2)))/y(x) + Derivative(y(x), x) cannot be solved by the factorable group method