12.3.12 problem 13

Internal problem ID [1589]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. separable equations. Section 2.2 Page 52
Problem number : 13
Date solved : Tuesday, March 04, 2025 at 12:41:09 PM
CAS classification : [_separable]

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

With initial conditions

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

✓ Maple. Time used: 0.374 (sec). Leaf size: 102

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

\[ y = \frac {\left (260-108 x^{2}-108 \sin \left (x \right )+12 \sqrt {441-390 x^{2}-390 \sin \left (x \right )+81 x^{4}+162 \sin \left (x \right ) x^{2}+81 \sin \left (x \right )^{2}}\right )^{{1}/{3}}}{6}+\frac {8}{3 \left (260-108 x^{2}-108 \sin \left (x \right )+12 \sqrt {441-390 x^{2}-390 \sin \left (x \right )+81 x^{4}+162 \sin \left (x \right ) x^{2}+81 \sin \left (x \right )^{2}}\right )^{{1}/{3}}}-\frac {2}{3} \]

✓ Mathematica. Time used: 2.426 (sec). Leaf size: 127

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

\[ y(x)\to \frac {2^{2/3} \left (-27 x^2+\sqrt {\left (27 x^2+27 \sin (x)-65\right )^2-256}-27 \sin (x)+65\right )^{2/3}-4 \sqrt [3]{-27 x^2+\sqrt {\left (27 x^2+27 \sin (x)-65\right )^2-256}-27 \sin (x)+65}+8 \sqrt [3]{2}}{6 \sqrt [3]{-27 x^2+\sqrt {\left (27 x^2+27 \sin (x)-65\right )^2-256}-27 \sin (x)+65}} \]

✓ Sympy. Time used: 122.116 (sec). Leaf size: 221

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

\[ y{\left (x \right )} = \frac {\sqrt [3]{\frac {27 x^{2}}{2} + \frac {3 \sqrt {3} \sqrt {27 x^{4} + 54 x^{2} \sin {\left (x \right )} - 130 x^{2} + 27 \sin ^{2}{\left (x \right )} - 130 \sin {\left (x \right )} + 147}}{2} + \frac {27 \sin {\left (x \right )}}{2} - \frac {65}{2}}}{6} - \frac {\sqrt {3} i \sqrt [3]{\frac {27 x^{2}}{2} + \frac {3 \sqrt {3} \sqrt {27 x^{4} + 54 x^{2} \sin {\left (x \right )} - 130 x^{2} + 27 \sin ^{2}{\left (x \right )} - 130 \sin {\left (x \right )} + 147}}{2} + \frac {27 \sin {\left (x \right )}}{2} - \frac {65}{2}}}{6} - \frac {2}{3} - \frac {8}{3 \left (-1 + \sqrt {3} i\right ) \sqrt [3]{\frac {27 x^{2}}{2} + \frac {3 \sqrt {3} \sqrt {27 x^{4} + 54 x^{2} \sin {\left (x \right )} - 130 x^{2} + 27 \sin ^{2}{\left (x \right )} - 130 \sin {\left (x \right )} + 147}}{2} + \frac {27 \sin {\left (x \right )}}{2} - \frac {65}{2}}} \]