56.2.10 problem 10
Internal
problem
ID
[8814]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
2.0
Problem
number
:
10
Date
solved
:
Sunday, March 30, 2025 at 01:40:40 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }-x y^{\prime }-x y-x^{3}&=0 \end{align*}
✓ Maple. Time used: 0.018 (sec). Leaf size: 201
ode:=diff(diff(y(x),x),x)-x*diff(y(x),x)-x*y(x)-x^3 = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {{\mathrm e}^{-x} \int x^{3} \left ({\mathrm e}^{x} \sqrt {\pi }\, \sqrt {2}+i {\mathrm e}^{-2-x -\frac {1}{2} x^{2}} \pi \,\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +2\right )}{2}\right ) \left (x +2\right )\right )d x \sqrt {2}\, \left (x +2\right )+i x \left (x^{2}+x +2\right ) \left (x +2\right ) \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +2\right )}{2}\right ) \sqrt {2}\, \pi \,{\mathrm e}^{-\frac {\left (x +2\right )^{2}}{2}}-2 \,{\mathrm e}^{\frac {\left (x +1\right )^{2}}{2}} \operatorname {erf}\left (\frac {\sqrt {2}\, \left (x +1\right )}{2}\right ) \sqrt {2}\, \pi -2 \pi ^{{3}/{2}} \left (x +2\right ) \left (i {\mathrm e}^{-\frac {3}{2}-x} \operatorname {erf}\left (\frac {\sqrt {2}\, \left (x +1\right )}{2}\right )-{\mathrm e}^{-2-x} c_1 \right ) \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +2\right )}{2}\right )-2 i {\mathrm e}^{\frac {x \left (x +2\right )}{2}} \sqrt {2}\, c_1 \pi +2 \sqrt {\pi }\, \left ({\mathrm e}^{-x} \left (x +2\right ) c_2 +x^{3}+x^{2}+2 x \right )}{2 \sqrt {\pi }}
\]
✓ Mathematica. Time used: 4.175 (sec). Leaf size: 453
ode=D[y[x],{x,2}]-x*D[y[x],x]-x*y[x]-x^3==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {1}{2} e^{-\frac {1}{2} (x+2)^2} \left (2 \sqrt {2} e^{\frac {x^2}{2}+x+2} (x+2) \int _1^x\left (\frac {e^{K[1]} K[1]^3}{\sqrt {2}}-\frac {1}{2} e^{-\frac {1}{2} K[1]^2-K[1]-2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {(K[1]+2)^2}}{\sqrt {2}}\right ) K[1]^3 \sqrt {(K[1]+2)^2}\right )dK[1]-2 \text {erf}\left (\frac {x+1}{\sqrt {2}}\right ) \left (\sqrt {2 \pi } e^{x^2+3 x+\frac {5}{2}}-\pi e^{\frac {1}{2} (x+1)^2} \sqrt {(x+2)^2} \text {erfi}\left (\frac {\sqrt {(x+2)^2}}{\sqrt {2}}\right )\right )-\sqrt {2 \pi } \sqrt {(x+2)^2} x^3 \text {erfi}\left (\frac {\sqrt {(x+2)^2}}{\sqrt {2}}\right )-\sqrt {2 \pi } \sqrt {(x+2)^2} x^2 \text {erfi}\left (\frac {\sqrt {(x+2)^2}}{\sqrt {2}}\right )-\sqrt {2 \pi } c_2 e^{\frac {x^2}{2}+x+2} \sqrt {(x+2)^2} \text {erfi}\left (\frac {\sqrt {(x+2)^2}}{\sqrt {2}}\right )-2 \sqrt {2 \pi } \sqrt {(x+2)^2} x \text {erfi}\left (\frac {\sqrt {(x+2)^2}}{\sqrt {2}}\right )+2 e^{\frac {1}{2} (x+2)^2} x^3+2 e^{\frac {1}{2} (x+2)^2} x^2+2 \sqrt {2} c_1 e^{\frac {x^2}{2}+x+2} x+4 \sqrt {2} c_1 e^{\frac {x^2}{2}+x+2}+2 c_2 e^{x^2+3 x+4}+4 e^{\frac {1}{2} (x+2)^2} x\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**3 - x*y(x) - x*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE x**2 + y(x) + Derivative(y(x), x) - Derivative(y(x), (x, 2))/x cannot be solved by the factorable group method