Internal problem ID [6396]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 13.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-a x y^{\prime }-b x y-x^{3} c=0} \end {gather*}
✓ Solution by Maple
Time used: 0.079 (sec). Leaf size: 524
dsolve(diff(y(x),x$2)-a*x*diff(y(x),x)-b*x*y(x)-c*x^3=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{-\frac {b x}{a}} \KummerM \left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) c_{2}+{\mathrm e}^{-\frac {b x}{a}} \KummerU \left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) c_{1}+\frac {2 a^{4} c \left (\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} \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}}}{\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 ) \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 \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 ) \KummerU \left (-\frac {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 )-\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} \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}}}{\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 ) \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 \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 ) \KummerU \left (-\frac {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 )\right ) {\mathrm e}^{-\frac {b x}{a}}}{b^{2}} \]
✓ Solution by Mathematica
Time used: 0.663 (sec). Leaf size: 569
DSolve[y''[x]-a*x*y'[x]-b*x*y[x]-c*x^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-\frac {b x}{a}} \left (\text {HermiteH}\left (\frac {b^2}{a^3},\frac {a^2 x+2 b}{\sqrt {2} a^{3/2}}\right ) \int _1^x\frac {a^4 c e^{\frac {b K[1]}{a}} \, _1F_1\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} \text {HermiteH}\left (\frac {b^2}{a^3}-1,\frac {K[1] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \, _1F_1\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}+\text {HermiteH}\left (\frac {b^2}{a^3},\frac {K[1] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \, _1F_1\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]+\, _1F_1\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}} \text {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} \text {HermiteH}\left (\frac {b^2}{a^3}-1,\frac {K[2] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \, _1F_1\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}+\text {HermiteH}\left (\frac {b^2}{a^3},\frac {K[2] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \, _1F_1\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_1 \text {HermiteH}\left (\frac {b^2}{a^3},\frac {a^2 x+2 b}{\sqrt {2} a^{3/2}}\right )+c_2 \, _1F_1\left (-\frac {b^2}{2 a^3};\frac {1}{2};\frac {\left (x a^2+2 b\right )^2}{2 a^3}\right )\right ) \\ \end{align*}