12.10.34 problem 34

Internal problem ID [1838]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page 262
Problem number : 34
Date solved : Saturday, March 29, 2025 at 11:41:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

With initial conditions

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 16

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

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

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 21

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

\[ y(x)\to -\frac {x^2}{2}+\frac {1}{2 x^2}+x \]

✓ Sympy. Time used: 0.238 (sec). Leaf size: 15

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

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