89.6.21 problem 21

Internal problem ID [24404]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Miscellaneous Exercises at page 45
Problem number : 21
Date solved : Thursday, October 02, 2025 at 10:25:36 PM
CAS classification : [_linear]

\begin{align*} 1+4 y x -4 x^{2} y+\left (-x^{3}+x^{2}\right ) y^{\prime }&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 19

ode:=1+4*x*y(x)-4*x^2*y(x)+(-x^3+x^2)*diff(y(x),x) = 0; 
ic:=[y(2) = 1/4]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 24

ode=( 1+4*x*y[x] - 4*x^2 * y[x])+(x^2-x^3 )*D[y[x],x]==0; 
ic={y[2]==1/4}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x (x+2)+2 \log (x-1)}{2 x^4} \end{align*}

✓ Sympy. Time used: 0.182 (sec). Leaf size: 20

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

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