61.31.21 problem 202
Internal
problem
ID
[12623]
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
:
Monday, March 31, 2025 at 06:06:54 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*}
✓ Maple. Time used: 0.105 (sec). Leaf size: 63
ode:=2*x*(a*x^2+b*x+c)*diff(diff(y(x),x),x)+(a*x^2-c)*diff(y(x),x)+lambda*x^2*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \left (c_1 \,{\mathrm e}^{i \sqrt {2}\, \sqrt {\lambda }\, \int \frac {\sqrt {x}}{\sqrt {a \,x^{2}+b x +c}}d x}+c_2 \right ) {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {\lambda }\, \int \frac {\sqrt {x}}{\sqrt {a \,x^{2}+b x +c}}d x}{2}}
\]
✓ Mathematica. Time used: 122.817 (sec). Leaf size: 501
ode=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;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to c_1 \cosh \left (\frac {\sqrt {\lambda } \left (\sqrt {b^2-4 a c}-b\right ) \sqrt {\sqrt {b^2-4 a c}+2 a x+b} \sqrt {\frac {2 a x}{b-\sqrt {b^2-4 a c}}+1} \left (E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{2 a^{3/2} \sqrt {x (a x+b)+c}}\right )+i c_2 \sinh \left (\frac {\sqrt {\lambda } \left (\sqrt {b^2-4 a c}-b\right ) \sqrt {\sqrt {b^2-4 a c}+2 a x+b} \sqrt {\frac {2 a x}{b-\sqrt {b^2-4 a c}}+1} \left (E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{2 a^{3/2} \sqrt {x (a x+b)+c}}\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(lambda_*x**2*y(x) + 2*x*(a*x**2 + b*x + c)*Derivative(y(x), (x, 2)) + (a*x**2 - c)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
False