61.31.21 problem 202
Internal
problem
ID
[12702]
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
Problem
number
:
202
Date
solved
:
Tuesday, January 28, 2025 at 03:49:10 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\begin{align*} 2 x \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (a \,x^{2}-c \right ) y^{\prime }+\lambda \,x^{2} y&=0 \end{align*}
✓ Solution by Maple
Time used: 11.981 (sec). Leaf size: 63
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)
\[
y = c_{1} {\mathrm e}^{\frac {i \sqrt {2}\, \sqrt {\lambda }\, \left (\int \frac {\sqrt {x}}{\sqrt {a \,x^{2}+b x +c}}d x \right )}{2}}+c_{2} {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {\lambda }\, \left (\int \frac {\sqrt {x}}{\sqrt {a \,x^{2}+b x +c}}d x \right )}{2}}
\]
✓ Solution by Mathematica
Time used: 122.817 (sec). Leaf size: 501
DSolve[2*x*(a*x^2+b*x+c)*D[y[x],{x,2}]+(a*x^2-c)*D[y[x],x]+\[Lambda]*x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to c_1 \cosh \left (\frac {\sqrt {\lambda } \left (\sqrt {b^2-4 a c}-b\right ) \sqrt {\sqrt {b^2-4 a c}+2 a x+b} \sqrt {\frac {2 a x}{b-\sqrt {b^2-4 a c}}+1} \left (E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{2 a^{3/2} \sqrt {x (a x+b)+c}}\right )+i c_2 \sinh \left (\frac {\sqrt {\lambda } \left (\sqrt {b^2-4 a c}-b\right ) \sqrt {\sqrt {b^2-4 a c}+2 a x+b} \sqrt {\frac {2 a x}{b-\sqrt {b^2-4 a c}}+1} \left (E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{2 a^{3/2} \sqrt {x (a x+b)+c}}\right )
\]