48.2.8 problem Example 3.26
Internal
problem
ID
[7524]
Book
:
THEORY
OF
DIFFERENTIAL
EQUATIONS
IN
ENGINEERING
AND
MECHANICS.
K.T.
CHAU,
CRC
Press.
Boca
Raton,
FL.
2018
Section
:
Chapter
3.
Ordinary
Differential
Equations.
Section
3.3
SECOND
ORDER
ODE.
Page
147
Problem
number
:
Example
3.26
Date
solved
:
Monday, January 27, 2025 at 03:04:29 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} p \,x^{2} u^{\prime \prime }+q x u^{\prime }+r u&=f \left (x \right ) \end{align*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 259
dsolve(p*x^2*diff(u(x),x$2)+q*x*diff(u(x),x)+r*u(x)=f(x),u(x), singsol=all)
\[
u = \frac {x^{\frac {-q +p +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} c_{2} \sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}+x^{-\frac {q -p +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} c_{1} \sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}+x^{\frac {-q +p +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} \left (\int x^{-\frac {3 p -q +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} f \left (x \right )d x \right )-x^{-\frac {q -p +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} \left (\int x^{\frac {-3 p +q +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} f \left (x \right )d x \right )}{\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}
\]
✓ Solution by Mathematica
Time used: 1.216 (sec). Leaf size: 342
DSolve[p*x^2*D[u[x],{x,2}]+q*x*D[u[x],x]+r*u[x]==f[x],u[x],x,IncludeSingularSolutions -> True]
\[
u(x)\to x^{-\frac {\sqrt {p} \sqrt {r} \sqrt {\frac {p^2-2 p (q+2 r)+q^2}{p r}}-p+q}{2 p}} \left (x^{\frac {\sqrt {r} \sqrt {\frac {p^2-2 p (q+2 r)+q^2}{p r}}}{\sqrt {p}}} \int _1^x\frac {f(K[2]) K[2]^{\frac {-3 p-\sqrt {r} \sqrt {\frac {p^2-2 (q+2 r) p+q^2}{p r}} \sqrt {p}+q}{2 p}}}{\sqrt {p} \sqrt {r} \sqrt {\frac {p^2-2 (q+2 r) p+q^2}{p r}}}dK[2]+\int _1^x-\frac {f(K[1]) K[1]^{\frac {-3 p+\sqrt {r} \sqrt {\frac {p^2-2 (q+2 r) p+q^2}{p r}} \sqrt {p}+q}{2 p}}}{\sqrt {p} \sqrt {r} \sqrt {\frac {p^2-2 (q+2 r) p+q^2}{p r}}}dK[1]+c_2 x^{\frac {\sqrt {r} \sqrt {\frac {p^2-2 p (q+2 r)+q^2}{p r}}}{\sqrt {p}}}+c_1\right )
\]