44.4.10 problem 10

Internal problem ID [9145]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.5. Exact Equations. Page 20
Problem number : 10
Date solved : Tuesday, September 30, 2025 at 06:07:25 PM
CAS classification : [_exact, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 298

ode:=2*x*y(x)^3+cos(x)*y(x)+(3*x^2*y(x)^2+sin(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\left (12 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}+4 \sin \left (x \right )^{3}}-108 c_1 x \right )^{{2}/{3}}-12 \sin \left (x \right )}{6 x \left (12 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}+4 \sin \left (x \right )^{3}}-108 c_1 x \right )^{{1}/{3}}} \\ y &= -\frac {i \sqrt {3}\, \left (12 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}+4 \sin \left (x \right )^{3}}-108 c_1 x \right )^{{2}/{3}}+12 i \sin \left (x \right ) \sqrt {3}+\left (12 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}+4 \sin \left (x \right )^{3}}-108 c_1 x \right )^{{2}/{3}}-12 \sin \left (x \right )}{12 x \left (12 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}+4 \sin \left (x \right )^{3}}-108 c_1 x \right )^{{1}/{3}}} \\ y &= \frac {i \sqrt {3}\, \left (12 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}+4 \sin \left (x \right )^{3}}-108 c_1 x \right )^{{2}/{3}}+12 i \sin \left (x \right ) \sqrt {3}-\left (12 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}+4 \sin \left (x \right )^{3}}-108 c_1 x \right )^{{2}/{3}}+12 \sin \left (x \right )}{12 x \left (12 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}+4 \sin \left (x \right )^{3}}-108 c_1 x \right )^{{1}/{3}}} \\ \end{align*}

✓ Mathematica. Time used: 0.164 (sec). Leaf size: 70

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (3 x^2 K[2]^2+\sin (x)-\int _1^x\left (6 K[1] K[2]^2+\cos (K[1])\right )dK[1]\right )dK[2]+\int _1^x\left (2 K[1] y(x)^3+\cos (K[1]) y(x)\right )dK[1]=c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out