20.4.23 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 : Monday, January 27, 2025 at 07:52:29 AM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.060 (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: 5.488 (sec). Leaf size: 227

DSolve[D[y[x],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*}