69.20.8 problem 643

Internal problem ID [18429]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 643
Date solved : Thursday, October 02, 2025 at 03:11:49 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }-3 y&=5 x^{4} \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {1}{x} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 18

ode:=x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)-3*y(x) = 5*x^4; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_2 \,x^{4}+x^{5}+c_1}{x} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 21

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

\begin{align*} y(x)&\to \frac {x^5+c_2 x^4+c_1}{x} \end{align*}

✓ Sympy. Time used: 0.202 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = \frac {C_{1} + x^{4} \left (C_{2} + x\right )}{x} \]