Internal
problem
ID
[8243]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
2.0
Problem
number
:
13
Date
solved
:
Sunday, November 10, 2024 at 09:05:02 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }-a x y^{\prime }-b x y-c \,x^{3}&=0 \end{align*}
Maple dsolve solution
Solving time : 0.037
(sec)
Leaf size : 517
dsolve(diff(diff(y(x),x),x)-a*x*diff(y(x),x)-b*x*y(x)-c*x^3 = 0,
y(x),singsol=all)
\[
y = \frac {{\mathrm e}^{-\frac {b x}{a}} \left (2 \operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) \left (\int -\frac {\left (a^{2} x +2 b \right ) x^{3} \operatorname {KummerM}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) {\mathrm e}^{\frac {b x}{a}}}{\operatorname {KummerM}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) \left (a^{3}-b^{2}\right ) \operatorname {KummerU}\left (\frac {2 a^{3}-b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right )-2 \operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) \operatorname {KummerM}\left (\frac {2 a^{3}-b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) a^{3}}d x \right ) a^{4} c -2 \operatorname {KummerM}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) \left (\int -\frac {\left (a^{2} x +2 b \right ) x^{3} \operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) {\mathrm e}^{\frac {b x}{a}}}{\operatorname {KummerM}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) \left (a^{3}-b^{2}\right ) \operatorname {KummerU}\left (\frac {2 a^{3}-b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right )-2 \operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) \operatorname {KummerM}\left (\frac {2 a^{3}-b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) a^{3}}d x \right ) a^{4} c +b^{2} c_{1} \operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right )+b^{2} c_{2} \operatorname {KummerM}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right )\right )}{b^{2}}
\]
Mathematica DSolve solution
Solving time : 2.885
(sec)
Leaf size : 569
DSolve[{D[y[x],{x,2}]-a*x*D[y[x],x]-b*x*y[x]-c*x^3==0,{}},
y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to e^{-\frac {b x}{a}} \left (\operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {x a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \int _1^x\frac {a^4 c e^{\frac {b K[1]}{a}} \operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (K[1] a^2+2 b\right )^2}{2 a^3}\right ) K[1]^3}{b^2 \left (\sqrt {2} \operatorname {HermiteH}\left (\frac {b^2}{a^3}-1,\frac {K[1] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (K[1] a^2+2 b\right )^2}{2 a^3}\right ) a^{3/2}+\operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {K[1] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \operatorname {Hypergeometric1F1}\left (1-\frac {b^2}{2 a^3},\frac {3}{2},\frac {\left (K[1] a^2+2 b\right )^2}{2 a^3}\right ) \left (K[1] a^2+2 b\right )\right )}dK[1]+\operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (x a^2+2 b\right )^2}{2 a^3}\right ) \int _1^x-\frac {a^4 c e^{\frac {b K[2]}{a}} \operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {K[2] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) K[2]^3}{b^2 \left (\sqrt {2} \operatorname {HermiteH}\left (\frac {b^2}{a^3}-1,\frac {K[2] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (K[2] a^2+2 b\right )^2}{2 a^3}\right ) a^{3/2}+\operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {K[2] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \operatorname {Hypergeometric1F1}\left (1-\frac {b^2}{2 a^3},\frac {3}{2},\frac {\left (K[2] a^2+2 b\right )^2}{2 a^3}\right ) \left (K[2] a^2+2 b\right )\right )}dK[2]+c_2 \operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (x a^2+2 b\right )^2}{2 a^3}\right )+c_1 \operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {x a^2+2 b}{\sqrt {2} a^{3/2}}\right )\right )
\]