61.29.30 problem 139
Internal
problem
ID
[12639]
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
:
Tuesday, January 28, 2025 at 08:03:02 PM
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*}
✓ Solution by Maple
Time used: 0.234 (sec). Leaf size: 150
dsolve(a__2*x^2*diff(y(x),x$2)+(a__1*x^2+b__1*x)*diff(y(x),x)+(a__0*x^2+b__0*x+c__0)*y(x)=0,y(x), singsol=all)
\[
y = {\mathrm e}^{-\frac {a_{1} x}{2 a_{2}}} x^{-\frac {b_{1}}{2 a_{2}}} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {b_{1} a_{1} -2 a_{2} b_{0}}{2 a_{2} \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}}, \frac {\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, x}{a_{2}}\right )+c_{2} \operatorname {WhittakerW}\left (-\frac {b_{1} a_{1} -2 a_{2} b_{0}}{2 a_{2} \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}}, \frac {\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, x}{a_{2}}\right )\right )
\]
✓ Solution by Mathematica
Time used: 1.136 (sec). Leaf size: 280
DSolve[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,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \left (c_1 \operatorname {HypergeometricU}\left (\frac {-\frac {2 \text {b0} \text {a2}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\text {a2}+\frac {\text {a1} \text {b1}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{2 \text {a2}},\frac {\text {a2}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{\text {a2}},\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} x}{\text {a2}}\right )+c_2 L_{-\frac {-\frac {2 \text {b0} \text {a2}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\text {a2}+\frac {\text {a1} \text {b1}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{2 \text {a2}}}^{\frac {\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{\text {a2}}}\left (\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} x}{\text {a2}}\right )\right ) \exp \left (\int _1^x\frac {\text {a2}-\text {b1}-\text {a1} K[1]-\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} K[1]+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{2 \text {a2} K[1]}dK[1]\right )
\]