88.23.19 problem 22

Internal problem ID [24193]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 5. Special Techniques for Linear Equations. Miscellaneous Exercises at page 162
Problem number : 22
Date solved : Thursday, October 02, 2025 at 10:00:40 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }+a x y^{\prime }+b y&=f \left (x \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 157

ode:=x^2*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+b*y(x) = f(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\left (\sqrt {a^{2}-2 a -4 b +1}\, c_1 -\int \frac {x^{\frac {a}{2}} x^{\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}} f \left (x \right )}{x^{{3}/{2}}}d x \right ) x^{-\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}}+x^{\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}} \left (\sqrt {a^{2}-2 a -4 b +1}\, c_2 +\int \frac {x^{\frac {a}{2}} x^{-\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}} f \left (x \right )}{x^{{3}/{2}}}d x \right )\right ) x^{-\frac {a}{2}} \sqrt {x}}{\sqrt {a^{2}-2 a -4 b +1}} \]

✓ Mathematica. Time used: 0.229 (sec). Leaf size: 180

ode=x^2*D[y[x],{x,2}]+a*x*D[y[x],{x,1}]+b*y[x]==f[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 b+1}-a+1\right )} \left (x^{\sqrt {a^2-2 a-4 b+1}} \int _1^x\frac {f(K[2]) K[2]^{\frac {1}{2} \left (a-\sqrt {a^2-2 a-4 b+1}-3\right )}}{\sqrt {a^2-2 a-4 b+1}}dK[2]+\int _1^x-\frac {f(K[1]) K[1]^{\frac {1}{2} \left (a+\sqrt {a^2-2 a-4 b+1}-3\right )}}{\sqrt {a^2-2 a-4 b+1}}dK[1]+c_2 x^{\sqrt {a^2-2 a-4 b+1}}+c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
f = Function("f") 
ode = Eq(a*x*Derivative(y(x), x) + b*y(x) + x**2*Derivative(y(x), (x, 2)) - f(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out