Internal problem ID [2178]
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]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {y \tan \relax (x )}{2}-2 y^{3} \sin \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.037 (sec). Leaf size: 66
dsolve(diff(y(x),x)+1/2*tan(x)*y(x)=2*y(x)^3*sin(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {-\left (2 \left (\sin ^{2}\relax (x )\right )-c_{1}\right ) \cos \relax (x )}}{2 \left (\sin ^{2}\relax (x )\right )-c_{1}} \\ y \relax (x ) = -\frac {\sqrt {-\left (2 \left (\sin ^{2}\relax (x )\right )-c_{1}\right ) \cos \relax (x )}}{2 \left (\sin ^{2}\relax (x )\right )-c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.899 (sec). Leaf size: 215
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 (\sqrt {2 \pi } \left (\text {Erfi}\left (\frac {1+i \text {Tan} x}{\sqrt {2} \sqrt {\text {Tan}}}\right )-i \text {Erf}\left (\frac {\text {Tan} x+i}{\sqrt {2} \sqrt {\text {Tan}}}\right )\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 (\sqrt {2 \pi } \left (\text {Erfi}\left (\frac {1+i \text {Tan} x}{\sqrt {2} \sqrt {\text {Tan}}}\right )-i \text {Erf}\left (\frac {\text {Tan} x+i}{\sqrt {2} \sqrt {\text {Tan}}}\right )\right )+c_1 e^{\left .\frac {1}{2}\right /\text {Tan}} \sqrt {\text {Tan}}\right )}} \\ y(x)\to 0 \\ \end{align*}