Internal problem ID [11026]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-6 Equation of form
\((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 202.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {2 y^{\prime \prime } x \left (a \,x^{2}+b x +c \right )+\left (a \,x^{2}-c \right ) y^{\prime }+\lambda \,x^{2} y=0} \]
✓ Solution by Maple
Time used: 7.125 (sec). Leaf size: 65
dsolve(2*x*(a*x^2+b*x+c)*diff(y(x),x$2)+(a*x^2-c)*diff(y(x),x)+lambda*x^2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {i \sqrt {2}\, \sqrt {\lambda }\, \left (\int \frac {\sqrt {x}}{\sqrt {a \,x^{2}+b x +c}}d x \right )}{2}}+c_{2} {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {\lambda }\, \left (\int \frac {\sqrt {x}}{\sqrt {a \,x^{2}+b x +c}}d x \right )}{2}} \]
✓ Solution by Mathematica
Time used: 144.69 (sec). Leaf size: 501
DSolve[2*x*(a*x^2+b*x+c)*y''[x]+(a*x^2-c)*y'[x]+\[Lambda]*x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \cosh \left (\frac {\sqrt {\lambda } \left (\sqrt {b^2-4 a c}-b\right ) \sqrt {\sqrt {b^2-4 a c}+2 a x+b} \sqrt {\frac {2 a x}{b-\sqrt {b^2-4 a c}}+1} \left (E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{2 a^{3/2} \sqrt {x (a x+b)+c}}\right )+i c_2 \sinh \left (\frac {\sqrt {\lambda } \left (\sqrt {b^2-4 a c}-b\right ) \sqrt {\sqrt {b^2-4 a c}+2 a x+b} \sqrt {\frac {2 a x}{b-\sqrt {b^2-4 a c}}+1} \left (E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{2 a^{3/2} \sqrt {x (a x+b)+c}}\right ) \]