14.4.23 problem Problem 39

Internal problem ID [3658]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.8, Change of Variables. page 79
Problem number : Problem 39
Date solved : Tuesday, September 30, 2025 at 06:53:21 AM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 56

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

\begin{align*} y &= -\frac {\sqrt {\left (-2 \sin \left (x \right )^{2}+c_1 \right ) \cos \left (x \right )}}{-2 \sin \left (x \right )^{2}+c_1} \\ y &= \frac {\sqrt {\left (-2 \sin \left (x \right )^{2}+c_1 \right ) \cos \left (x \right )}}{-2 \sin \left (x \right )^{2}+c_1} \\ \end{align*}

✓ Mathematica. Time used: 4.802 (sec). Leaf size: 227

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

\begin{align*} y(x)&\to -\frac {e^{\left .\frac {1}{4}\right /\text {Tan}} \sqrt [4]{\text {Tan}}}{\sqrt {e^{\frac {\text {Tan} x^2}{2}} \left (-i \sqrt {2 \pi } \text {erf}\left (\frac {\text {Tan} x+i}{\sqrt {2} \sqrt {\text {Tan}}}\right )+\sqrt {2 \pi } \text {erfi}\left (\frac {1+i \text {Tan} x}{\sqrt {2} \sqrt {\text {Tan}}}\right )+c_1 e^{\left .\frac {1}{2}\right /\text {Tan}} \sqrt {\text {Tan}}\right )}}\\ y(x)&\to \frac {e^{\left .\frac {1}{4}\right /\text {Tan}} \sqrt [4]{\text {Tan}}}{\sqrt {e^{\frac {\text {Tan} x^2}{2}} \left (-i \sqrt {2 \pi } \text {erf}\left (\frac {\text {Tan} x+i}{\sqrt {2} \sqrt {\text {Tan}}}\right )+\sqrt {2 \pi } \text {erfi}\left (\frac {1+i \text {Tan} x}{\sqrt {2} \sqrt {\text {Tan}}}\right )+c_1 e^{\left .\frac {1}{2}\right /\text {Tan}} \sqrt {\text {Tan}}\right )}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.866 (sec). Leaf size: 32

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

\[ \left [ y{\left (x \right )} = - \sqrt {\frac {\cos {\left (x \right )}}{C_{1} + \cos {\left (2 x \right )}}}, \ y{\left (x \right )} = \sqrt {\frac {\cos {\left (x \right )}}{C_{1} + \cos {\left (2 x \right )}}}\right ] \]