Internal problem ID [5102]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {p \,x^{2} u^{\prime \prime }+q x u^{\prime }+r u-f \left (x \right )=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 222
dsolve(p*x^2*diff(u(x),x$2)+q*x*diff(u(x),x)+r*u(x)=f(x),u(x), singsol=all)
\[ u \left (x \right ) = x^{\frac {-q +p +\sqrt {p^{2}-2 q p -4 r p +q^{2}}}{2 p}} c_{2} +x^{-\frac {q -p +\sqrt {p^{2}-2 q p -4 r p +q^{2}}}{2 p}} c_{1} +\frac {-x^{-\frac {q -p +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} \left (\int x^{\frac {\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}-3 p +q}{2 p}} f \left (x \right )d x \right )+x^{\frac {-q +p +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} \left (\int x^{-\frac {\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}+3 p -q}{2 p}} f \left (x \right )d x \right )}{\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}} \]
✓ Solution by Mathematica
Time used: 0.65 (sec). Leaf size: 267
DSolve[p*x^2*u''[x]+q*x*u'[x]+r*u[x]==f[x],u[x],x,IncludeSingularSolutions -> True]
\begin{align*} u(x)\to x^{-\frac {\sqrt {p} \sqrt {r} \sqrt {\frac {(p-q)^2}{p r}-4}-p+q}{2 p}} \left (x^{\frac {\sqrt {r} \sqrt {\frac {(p-q)^2}{p r}-4}}{\sqrt {p}}} \left (\int _1^x\frac {f(K[2]) K[2]^{\frac {-3 p-\sqrt {\frac {(p-q)^2}{p r}-4} \sqrt {r} \sqrt {p}+q}{2 p}}}{\sqrt {p} \sqrt {\frac {(p-q)^2}{p r}-4} \sqrt {r}}dK[2]+c_2\right )+\int _1^x-\frac {f(K[1]) K[1]^{\frac {-3 p+\sqrt {\frac {(p-q)^2}{p r}-4} \sqrt {r} \sqrt {p}+q}{2 p}}}{\sqrt {p} \sqrt {\frac {(p-q)^2}{p r}-4} \sqrt {r}}dK[1]+c_1\right ) \\ \end{align*}