60.3.291 problem 1296
Internal
problem
ID
[11301]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1296
Date
solved
:
Tuesday, January 28, 2025 at 05:58:04 PM
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*}
✓ Solution by Maple
Time used: 0.191 (sec). Leaf size: 150
dsolve(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,y(x), singsol=all)
\[
y = {\mathrm e}^{-\frac {\operatorname {a1} x}{2 \operatorname {a2}}} x^{-\frac {\operatorname {b1}}{2 \operatorname {a2}}} \left (\operatorname {WhittakerM}\left (-\frac {\operatorname {a1} \operatorname {b1} -2 \operatorname {a2} \operatorname {b0}}{2 \operatorname {a2} \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}}, \frac {\sqrt {\operatorname {a2}^{2}+\left (-2 \operatorname {b1} -4 \operatorname {c0} \right ) \operatorname {a2} +\operatorname {b1}^{2}}}{2 \operatorname {a2}}, \frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, x}{\operatorname {a2}}\right ) c_{1} +c_{2} \operatorname {WhittakerW}\left (-\frac {\operatorname {a1} \operatorname {b1} -2 \operatorname {a2} \operatorname {b0}}{2 \operatorname {a2} \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}}, \frac {\sqrt {\operatorname {a2}^{2}+\left (-2 \operatorname {b1} -4 \operatorname {c0} \right ) \operatorname {a2} +\operatorname {b1}^{2}}}{2 \operatorname {a2}}, \frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, x}{\operatorname {a2}}\right )\right )
\]
✓ Solution by Mathematica
Time used: 1.166 (sec). Leaf size: 280
DSolve[(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,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \left (c_1 \operatorname {HypergeometricU}\left (\frac {-\frac {2 \text {b0} \text {a2}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\text {a2}+\frac {\text {a1} \text {b1}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{2 \text {a2}},\frac {\text {a2}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{\text {a2}},\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} x}{\text {a2}}\right )+c_2 L_{-\frac {-\frac {2 \text {b0} \text {a2}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\text {a2}+\frac {\text {a1} \text {b1}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{2 \text {a2}}}^{\frac {\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{\text {a2}}}\left (\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} x}{\text {a2}}\right )\right ) \exp \left (\int _1^x\frac {\text {a2}-\text {b1}-\text {a1} K[1]-\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} K[1]+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{2 \text {a2} K[1]}dK[1]\right )
\]