58.13.25 problem 25

Internal problem ID [14836]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.5. The Cauchy-Euler Equation. Exercises page 169
Problem number : 25
Date solved : Thursday, October 02, 2025 at 09:55:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+2 x y^{\prime }-6 y&=10 x^{2} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=1 \\ y^{\prime }\left (1\right )&=-6 \\ \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 22

ode:=x^2*diff(diff(y(x),x),x)+2*x*diff(y(x),x)-6*y(x) = 10*x^2; 
ic:=[y(1) = 1, D(y)(1) = -6]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {2 x^{5} \ln \left (x \right )-x^{5}+2}{x^{3}} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 23

ode=x^2*D[y[x],{x,2}]+2*x*D[y[x],x]-6*y[x]==10*x^2; 
ic={y[1]==1,Derivative[1][y][1]==-6}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {-x^5+2 x^5 \log (x)+2}{x^3} \end{align*}

✓ Sympy. Time used: 0.180 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - 10*x**2 + 2*x*Derivative(y(x), x) - 6*y(x),0) 
ics = {y(1): 1, Subs(Derivative(y(x), x), x, 1): -6} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x^{5} \left (2 \log {\left (x \right )} - 1\right ) + 2}{x^{3}} \]