54.3.280 problem 1296
Internal
problem
ID
[12575]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1296
Date
solved
:
Friday, October 03, 2025 at 03:38:24 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \operatorname {a2} \,x^{2} y^{\prime \prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x \right ) y^{\prime }+\left (\operatorname {a0} \,x^{2}+\operatorname {b0} x +\operatorname {c0} \right ) y&=0 \end{align*}
✓ Maple. Time used: 0.108 (sec). Leaf size: 150
ode:=a2*x^2*diff(diff(y(x),x),x)+(a1*x^2+b1*x)*diff(y(x),x)+(a0*x^2+b0*x+c0)*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = {\mathrm e}^{-\frac {x \operatorname {a1}}{2 \operatorname {a2}}} x^{-\frac {\operatorname {b1}}{2 \operatorname {a2}}} \left (\operatorname {WhittakerM}\left (-\frac {\operatorname {a1} \operatorname {b1} -2 \operatorname {a2} \operatorname {b0}}{2 \operatorname {a2} \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}}, \frac {\sqrt {\operatorname {a2}^{2}+\left (-2 \operatorname {b1} -4 \operatorname {c0} \right ) \operatorname {a2} +\operatorname {b1}^{2}}}{2 \operatorname {a2}}, \frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, x}{\operatorname {a2}}\right ) c_1 +\operatorname {WhittakerW}\left (-\frac {\operatorname {a1} \operatorname {b1} -2 \operatorname {a2} \operatorname {b0}}{2 \operatorname {a2} \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}}, \frac {\sqrt {\operatorname {a2}^{2}+\left (-2 \operatorname {b1} -4 \operatorname {c0} \right ) \operatorname {a2} +\operatorname {b1}^{2}}}{2 \operatorname {a2}}, \frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, x}{\operatorname {a2}}\right ) c_2 \right )
\]
✓ Mathematica. Time used: 0.885 (sec). Leaf size: 280
ode=(c0 + b0*x + a0*x^2)*y[x] + (b1*x + a1*x^2)*D[y[x],x] + a2*x^2*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \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 ) \exp \left (\int _1^x\frac {\text {a2}-\text {b1}-\text {a1} K[1]-\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} K[1]+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{2 \text {a2} K[1]}dK[1]\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a0 = symbols("a0")
a1 = symbols("a1")
a2 = symbols("a2")
b0 = symbols("b0")
b1 = symbols("b1")
c0 = symbols("c0")
y = Function("y")
ode = Eq(a2*x**2*Derivative(y(x), (x, 2)) + (a1*x**2 + b1*x)*Derivative(y(x), x) + (a0*x**2 + b0*x + c0)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ValueError : Expected Expr or iterable but got None