89.3.22 problem 23

Internal problem ID [24320]
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. Exercises at page 34
Problem number : 23
Date solved : Thursday, October 02, 2025 at 10:17:55 PM
CAS classification : [_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class A`]]

\begin{align*} 3 \left (x^{2}-1\right ) y+\left (x^{3}+8 y-3 x \right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 32

ode:=3*(x^2-1)*y(x)+(x^3+8*y(x)-3*x)*diff(y(x),x) = 0; 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {x^{3}}{8}+\frac {3 x}{8}+\frac {\sqrt {x^{6}-6 x^{4}+9 x^{2}+64}}{8} \]

✓ Mathematica. Time used: 0.116 (sec). Leaf size: 37

ode=3*y[x]*( x^2-1 )+ ( x^3+8*y[x]-3*x)*D[y[x],x]==0; 
ic={y[0]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{8} \left (-x^3+\sqrt {x^6-6 x^4+9 x^2+64}+3 x\right ) \end{align*}

✓ Sympy. Time used: 0.937 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*(x**2 - 1)*y(x) + (x**3 - 3*x + 8*y(x))*Derivative(y(x), x),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {x \left (x^{2} - 3\right )}{8} + \frac {\sqrt {x^{6} - 6 x^{4} + 9 x^{2} + 64}}{8} \]