Internal problem ID [10865]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-2 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 41.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a \,x^{3}+2 b \right ) y^{\prime }+\left (a b \,x^{3}-a \,x^{2}+b^{2}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 85
dsolve(diff(y(x),x$2)+(a*x^3+2*b)*diff(y(x),x)+(a*b*x^3-a*x^2+b^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\frac {7 \,2^{\frac {1}{4}} c_{2} a \left (x^{4} a \right )^{\frac {3}{8}} \left (x^{4} a +3\right ) {\mathrm e}^{-\frac {x \left (a \,x^{3}+4 b \right )}{4}}}{8}+{\mathrm e}^{-\frac {x \left (a \,x^{3}+8 b \right )}{8}} \operatorname {WhittakerM}\left (\frac {3}{8}, \frac {7}{8}, \frac {x^{4} a}{4}\right ) c_{2} a^{2} x^{4}+{\mathrm e}^{-b x} c_{1} x^{\frac {5}{2}}}{x^{\frac {3}{2}}} \]
✓ Solution by Mathematica
Time used: 0.431 (sec). Leaf size: 51
DSolve[y''[x]+(a*x^3+2*b)*y'[x]+(a*b*x^3-a*x^2+b^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{8} e^{-b x} \left (8 c_1 x-\sqrt {2} c_2 \sqrt [4]{a x^4} \Gamma \left (-\frac {1}{4},\frac {a x^4}{4}\right )\right ) \]