42.2.8 problem Example 3.26

Internal problem ID [8810]
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 : Tuesday, September 30, 2025 at 05:52:45 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*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 259

ode:=p*x^2*diff(diff(u(x),x),x)+q*x*diff(u(x),x)+r*u(x) = f(x); 
dsolve(ode,u(x), singsol=all);
 

\[ u = \frac {x^{\frac {-q +p +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} c_2 \sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}+x^{-\frac {q -p +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} c_1 \sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}+x^{\frac {-q +p +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} \int x^{-\frac {3 p -q +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} f \left (x \right )d x -\int x^{\frac {-3 p +q +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} f \left (x \right )d x x^{-\frac {q -p +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}}}{\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}} \]

✓ Mathematica. Time used: 0.7 (sec). Leaf size: 342

ode=p*x^2*D[u[x],{x,2}]+q*x*D[u[x],x]+r*u[x]==f[x]; 
ic={}; 
DSolve[{ode,ic},u[x],x,IncludeSingularSolutions->True]
 

\begin{align*} 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 ) \end{align*}

✓ Sympy. Time used: 150.045 (sec). Leaf size: 3026

from sympy import * 
x = symbols("x") 
p = symbols("p") 
q = symbols("q") 
r = symbols("r") 
u = Function("u") 
f = Function("f") 
ode = Eq(p*x**2*Derivative(u(x), (x, 2)) + q*x*Derivative(u(x), x) + r*u(x) - f(x),0) 
ics = {} 
dsolve(ode,func=u(x),ics=ics)
 

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