6.9.13 problem 13

Internal problem ID [1769]
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 : 13
Date solved : Tuesday, September 30, 2025 at 05:19:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y&=4 x^{4} \end{align*}

Using reduction of order method given that one solution is

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 17

ode:=x^2*diff(diff(y(x),x),x)-3*x*diff(y(x),x)+4*y(x) = 4*x^4; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{2} \left (\ln \left (x \right ) c_1 +x^{2}+c_2 \right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 21

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

\begin{align*} y(x)&\to x^2 \left (x^2+2 c_2 \log (x)+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.149 (sec). Leaf size: 15

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

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