6.9.31 problem 31

Internal problem ID [1787]
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 : 31
Date solved : Tuesday, September 30, 2025 at 05:19:19 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*}

With initial conditions

\begin{align*} y \left (-1\right )&=7 \\ y^{\prime }\left (-1\right )&=-8 \\ \end{align*}

✓ Maple. Time used: 0.036 (sec). Leaf size: 20

ode:=x^2*diff(diff(y(x),x),x)-3*x*diff(y(x),x)+4*y(x) = 4*x^4; 
ic:=[y(-1) = 7, D(y)(-1) = -8]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \left (8 i \pi +x^{2}-8 \ln \left (x \right )+6\right ) x^{2} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 37

ode=x^2*D[y[x],{x,2}]-3*x*D[y[x],x]+4*y[x]==4*x^2; 
ic={y[-1]==7,Derivative[1][y][-1]==8}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x^2 \left (2 \log ^2(x)+(-22-4 i \pi ) \log (x)-2 \pi ^2+22 i \pi +7\right ) \end{align*}

✓ Sympy. Time used: 0.161 (sec). Leaf size: 20

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 = {y(-1): 7, Subs(Derivative(y(x), x), x, -1): -8} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x^{2} \left (x^{2} - 8 \log {\left (x \right )} + 6 + 8 i \pi \right ) \]