6.9.34 problem 34

Internal problem ID [1790]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 34
Date solved : Tuesday, September 30, 2025 at 05:19:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=x \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&={\frac {5}{4}} \\ y^{\prime }\left (1\right )&={\frac {3}{2}} \\ \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 11

ode:=x^2*diff(diff(y(x),x),x)+2*x*diff(y(x),x)-2*y(x) = x^2; 
ic:=[y(1) = 5/4, D(y)(1) = 3/2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = x +\frac {1}{4} x^{2} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 13

ode=x^2*D[y[x],{x,2}]+2*x*D[y[x],x]-2*y[x]==x^2; 
ic={y[1]==5/4,Derivative[1][y][1]==3/2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{4} x (x+4) \end{align*}

✓ Sympy. Time used: 0.162 (sec). Leaf size: 8

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x**2 + 2*x*Derivative(y(x), x) - 2*y(x),0) 
ics = {y(1): 5/4, Subs(Derivative(y(x), x), x, 1): 3/2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x^{2}}{4} + x \]