60.3.296 problem 1302
Internal
problem
ID
[11306]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1302
Date
solved
:
Tuesday, January 28, 2025 at 05:58:07 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \operatorname {A2} \left (a x +b \right )^{2} y^{\prime \prime }+\operatorname {A1} \left (a x +b \right ) y^{\prime }+\operatorname {A0} \left (a x +b \right ) y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.127 (sec). Leaf size: 98
dsolve(A2*(a*x+b)^2*diff(diff(y(x),x),x)+A1*(a*x+b)*diff(y(x),x)+A0*(a*x+b)*y(x)=0,y(x), singsol=all)
\[
y = \left (a x +b \right )^{-\frac {-a \operatorname {A2} +\operatorname {A1}}{2 a \operatorname {A2}}} \left (\operatorname {BesselJ}\left (\frac {a \operatorname {A2} -\operatorname {A1}}{a \operatorname {A2}}, 2 \sqrt {\operatorname {A0}}\, \sqrt {\frac {a x +b}{a^{2} \operatorname {A2}}}\right ) c_{1} +\operatorname {BesselY}\left (\frac {a \operatorname {A2} -\operatorname {A1}}{a \operatorname {A2}}, 2 \sqrt {\operatorname {A0}}\, \sqrt {\frac {a x +b}{a^{2} \operatorname {A2}}}\right ) c_{2} \right )
\]
✓ Solution by Mathematica
Time used: 0.152 (sec). Leaf size: 165
DSolve[A0*(b + a*x)*y[x] + A1*(b + a*x)*D[y[x],x] + A2*(b + a*x)^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to (-1)^{-\frac {\text {A1}}{a \text {A2}}} \left (\frac {b}{a}+x\right )^{\frac {\text {A1}}{2 a \text {A2}}} (\text {A2} (a x+b))^{-\frac {\text {A1}}{2 a \text {A2}}} \left (-\frac {\text {A0} (a x+b)}{a^2 \text {A2}}\right )^{\frac {1}{2}-\frac {\text {A1}}{2 a \text {A2}}} \left (c_1 (-1)^{\frac {\text {A1}}{a \text {A2}}} \operatorname {BesselI}\left (\frac {\text {A1}}{a \text {A2}}-1,2 \sqrt {-\frac {\text {A0} (b+a x)}{a^2 \text {A2}}}\right )-c_2 K_{\frac {\text {A1}}{a \text {A2}}-1}\left (2 \sqrt {-\frac {\text {A0} (b+a x)}{a^2 \text {A2}}}\right )\right )
\]