55.30.30 problem 139
Internal
problem
ID
[13912]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.2-4
Problem
number
:
139
Date
solved
:
Friday, October 03, 2025 at 06:55:39 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} a_{2} x^{2} y^{\prime \prime }+\left (a_{1} x^{2}+b_{1} x \right ) y^{\prime }+\left (a_{0} x^{2}+b_{0} x +c_{0} \right ) y&=0 \end{align*}
✓ Maple. Time used: 0.107 (sec). Leaf size: 150
ode:=a__2*x^2*diff(diff(y(x),x),x)+(a__1*x^2+b__1*x)*diff(y(x),x)+(a__0*x^2+b__0*x+c__0)*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = {\mathrm e}^{-\frac {x a_{1}}{2 a_{2}}} x^{-\frac {b_{1}}{2 a_{2}}} \left (\operatorname {WhittakerM}\left (-\frac {a_{1} b_{1} -2 a_{2} b_{0}}{2 a_{2} \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}}, \frac {\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, x}{a_{2}}\right ) c_1 +\operatorname {WhittakerW}\left (-\frac {a_{1} b_{1} -2 a_{2} b_{0}}{2 a_{2} \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}}, \frac {\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, x}{a_{2}}\right ) c_2 \right )
\]
✓ Mathematica. Time used: 0.3 (sec). Leaf size: 272
ode=a2*x^2*D[y[x],{x,2}]+(a1*x^2+b1*x)*D[y[x],x]+(a0*x^2+b0*x+c0)*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to e^{-\frac {x \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}+\text {a1}\right )}{2 \text {a2}}} x^{\frac {\sqrt {\text {a2}^2-2 \text {a2} (\text {b1}+2 \text {c0})+\text {b1}^2}+\text {a2}-\text {b1}}{2 \text {a2}}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {-\frac {2 \text {b0} \text {a2}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\text {a2}+\frac {\text {a1} \text {b1}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{2 \text {a2}},\frac {\text {a2}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{\text {a2}},\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} x}{\text {a2}}\right )+c_2 L_{-\frac {-\frac {2 \text {b0} \text {a2}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\text {a2}+\frac {\text {a1} \text {b1}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{2 \text {a2}}}^{\frac {\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{\text {a2}}}\left (\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} x}{\text {a2}}\right )\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a__0 = symbols("a__0")
a__1 = symbols("a__1")
a__2 = symbols("a__2")
b__0 = symbols("b__0")
b__1 = symbols("b__1")
c__0 = symbols("c__0")
y = Function("y")
ode = Eq(a__2*x**2*Derivative(y(x), (x, 2)) + (a__1*x**2 + b__1*x)*Derivative(y(x), x) + (a__0*x**2 + b__0*x + c__0)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ValueError : Expected Expr or iterable but got None