Internal problem ID [2687]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
\[ \boxed {y^{\prime }+\frac {y \tan \left (x \right )}{2}-2 y^{3} \sin \left (x \right )=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 56
dsolve(diff(y(x),x)+1/2*tan(x)*y(x)=2*y(x)^3*sin(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\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 \left (x \right ) &= \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*}
✓ Solution by Mathematica
Time used: 5.32 (sec). Leaf size: 227
DSolve[y'[x]+1/2*Tan(x)*y[x]==2*y[x]^3*Sin[x],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*}