32.3.10 problem Exact Differential equations. Exercise 9.13, page 79

Internal problem ID [5808]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 9
Problem number : Exact Differential equations. Exercise 9.13, page 79
Date solved : Sunday, March 30, 2025 at 10:18:27 AM
CAS classification : [_exact]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 642

ode:=4*x^3-sin(x)+y(x)^3-(y(x)^2+1-3*x*y(x)^2)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 60.211 (sec). Leaf size: 682

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

\begin{align*} y(x)\to \frac {\sqrt [3]{2} \left (-27 x^6+18 x^5-3 x^4+\frac {1}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-27 x^2 \cos (x)+27 c_1 x^2+18 x \cos (x)-3 \cos (x)-18 c_1 x+3 c_1\right ){}^{2/3}+6 x-2}{2^{2/3} (3 x-1) \sqrt [3]{-27 x^6+18 x^5-3 x^4+\frac {1}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-27 x^2 \cos (x)+27 c_1 x^2+18 x \cos (x)-3 \cos (x)-18 c_1 x+3 c_1}} \\ y(x)\to \frac {9 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (-27 x^6+18 x^5-3 x^4+\frac {1}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-27 x^2 \cos (x)+27 c_1 x^2+18 x \cos (x)-3 \cos (x)-18 c_1 x+3 c_1\right ){}^{2/3}+2 \left (1+i \sqrt {3}\right ) (9-27 x)}{18\ 2^{2/3} (3 x-1) \sqrt [3]{-27 x^6+18 x^5-3 x^4+\frac {1}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-27 x^2 \cos (x)+27 c_1 x^2+18 x \cos (x)-3 \cos (x)-18 c_1 x+3 c_1}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right )}{2^{2/3} \sqrt [3]{-27 x^6+18 x^5-3 x^4+\frac {1}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-27 x^2 \cos (x)+27 c_1 x^2+18 x \cos (x)-3 \cos (x)-18 c_1 x+3 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-54 x^6+36 x^5-6 x^4+\frac {2}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-54 x^2 \cos (x)+54 c_1 x^2+36 x \cos (x)-6 \cos (x)-36 c_1 x+6 c_1}}{2\ 2^{2/3} (3 x-1)} \\ \end{align*}

✗ Sympy

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

Timed Out