2.13 problem 13

Internal problem ID [7797]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 13
Date solved : Tuesday, October 22, 2024 at 02:29:58 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

Solve

\begin{align*} y^{\prime \prime }-a x y^{\prime }-b x y-c \,x^{3}&=0 \end{align*}

2.13.1 Maple step by step solution

2.13.2 Maple trace

Methods for second order ODEs:
 

2.13.3 Maple dsolve solution

Solving time : 0.008 (sec)
Leaf size : 515

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 {KummerM}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (x \,a^{2}+2 b \right )^{2}}{2 a^{3}}\right ) \left (\int \frac {\left (x \,a^{2}+2 b \right ) x^{3} \operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (x \,a^{2}+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 (x \,a^{2}+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 (x \,a^{2}+2 b \right )^{2}}{2 a^{3}}\right )-2 \operatorname {KummerM}\left (\frac {2 a^{3}-b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (x \,a^{2}+2 b \right )^{2}}{2 a^{3}}\right ) \operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (x \,a^{2}+2 b \right )^{2}}{2 a^{3}}\right ) a^{3}}d x \right ) a^{4} c -2 \operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (x \,a^{2}+2 b \right )^{2}}{2 a^{3}}\right ) \left (\int \frac {\left (x \,a^{2}+2 b \right ) x^{3} \operatorname {KummerM}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (x \,a^{2}+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 (x \,a^{2}+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 (x \,a^{2}+2 b \right )^{2}}{2 a^{3}}\right )-2 \operatorname {KummerM}\left (\frac {2 a^{3}-b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (x \,a^{2}+2 b \right )^{2}}{2 a^{3}}\right ) \operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (x \,a^{2}+2 b \right )^{2}}{2 a^{3}}\right ) a^{3}}d x \right ) a^{4} c +\operatorname {KummerM}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (x \,a^{2}+2 b \right )^{2}}{2 a^{3}}\right ) c_2 \,b^{2}+\operatorname {KummerU}\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 \,b^{2}\right )}{b^{2}} \]

2.13.4 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]
 

\[ y(x)\to e^{-\frac {b x}{a}} \left (\operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {x a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \int _1^x\frac {a^4 c e^{\frac {b K[1]}{a}} \operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (K[1] a^2+2 b\right )^2}{2 a^3}\right ) K[1]^3}{b^2 \left (\sqrt {2} \operatorname {HermiteH}\left (\frac {b^2}{a^3}-1,\frac {K[1] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (K[1] a^2+2 b\right )^2}{2 a^3}\right ) a^{3/2}+\operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {K[1] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \operatorname {Hypergeometric1F1}\left (1-\frac {b^2}{2 a^3},\frac {3}{2},\frac {\left (K[1] a^2+2 b\right )^2}{2 a^3}\right ) \left (K[1] a^2+2 b\right )\right )}dK[1]+\operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (x a^2+2 b\right )^2}{2 a^3}\right ) \int _1^x-\frac {a^4 c e^{\frac {b K[2]}{a}} \operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {K[2] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) K[2]^3}{b^2 \left (\sqrt {2} \operatorname {HermiteH}\left (\frac {b^2}{a^3}-1,\frac {K[2] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (K[2] a^2+2 b\right )^2}{2 a^3}\right ) a^{3/2}+\operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {K[2] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \operatorname {Hypergeometric1F1}\left (1-\frac {b^2}{2 a^3},\frac {3}{2},\frac {\left (K[2] a^2+2 b\right )^2}{2 a^3}\right ) \left (K[2] a^2+2 b\right )\right )}dK[2]+c_2 \operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (x a^2+2 b\right )^2}{2 a^3}\right )+c_1 \operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {x a^2+2 b}{\sqrt {2} a^{3/2}}\right )\right ) \]