85.33.49 problem 49

Internal problem ID [22672]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 65
Problem number : 49
Date solved : Thursday, October 02, 2025 at 09:03:38 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class B`]]

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

With initial conditions

\begin{align*} y \left (0\right )&=-4 \\ \end{align*}

✓ Maple. Time used: 0.424 (sec). Leaf size: 99

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

\[ y = \frac {\sec \left (x \right ) \left (\left (i \sqrt {3}-1\right ) \left (-648 \cos \left (x \right )^{2}+36 \sqrt {324 \cos \left (x \right )^{2}+1}\, \cos \left (x \right )-1\right )^{{2}/{3}}-i \sqrt {3}-2 \left (-648 \cos \left (x \right )^{2}+36 \sqrt {324 \cos \left (x \right )^{2}+1}\, \cos \left (x \right )-1\right )^{{1}/{3}}-1\right )}{6 \left (-648 \cos \left (x \right )^{2}+36 \sqrt {324 \cos \left (x \right )^{2}+1}\, \cos \left (x \right )-1\right )^{{1}/{3}}} \]

✓ Mathematica. Time used: 66.235 (sec). Leaf size: 151

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

\begin{align*} y(x)&\to \frac {\left (i \sqrt {3} \left (-648 \cos ^2(x)+36 \sqrt {324 \cos ^4(x)+\cos ^2(x)}-1\right )^{2/3}-\left (-648 \cos ^2(x)+36 \sqrt {324 \cos ^4(x)+\cos ^2(x)}-1\right )^{2/3}-2 \sqrt [3]{-648 \cos ^2(x)+36 \sqrt {324 \cos ^4(x)+\cos ^2(x)}-1}-i \sqrt {3}-1\right ) \sec (x)}{6 \sqrt [3]{-648 \cos ^2(x)+36 \sqrt {324 \cos ^4(x)+\cos ^2(x)}-1}} \end{align*}

✓ Sympy. Time used: 49.626 (sec). Leaf size: 105

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

\[ y{\left (x \right )} = - \frac {\sqrt [3]{\frac {\sqrt {\frac {\left (1296 + \frac {2}{\cos ^{2}{\left (x \right )}}\right )^{2} - \frac {4}{\cos ^{4}{\left (x \right )}}}{\cos ^{2}{\left (x \right )}}}}{2} + \frac {648}{\cos {\left (x \right )}} + \frac {1}{\cos ^{3}{\left (x \right )}}}}{3} - \frac {1}{3 \cos {\left (x \right )}} - \frac {1}{3 \sqrt [3]{\frac {\sqrt {\frac {\left (1296 + \frac {2}{\cos ^{2}{\left (x \right )}}\right )^{2} - \frac {4}{\cos ^{4}{\left (x \right )}}}{\cos ^{2}{\left (x \right )}}}}{2} + \frac {648}{\cos {\left (x \right )}} + \frac {1}{\cos ^{3}{\left (x \right )}}} \cos ^{2}{\left (x \right )}} \]