60.1.350 problem 351

Internal problem ID [10364]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 351
Date solved : Tuesday, January 28, 2025 at 04:37:00 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime } \cos \left (y\right )+x \sin \left (y\right ) \cos \left (y\right )^{2}-\sin \left (y\right )^{3}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.073 (sec). Leaf size: 55

dsolve(diff(y(x),x)*cos(y(x))+x*sin(y(x))*cos(y(x))^2-sin(y(x))^3 = 0,y(x), singsol=all)
 

\begin{align*} y &= \arcsin \left (\frac {1}{\sqrt {1-\sqrt {\pi }\, {\mathrm e}^{x^{2}} \operatorname {erf}\left (x \right )-2 \,{\mathrm e}^{x^{2}} c_{1}}}\right ) \\ y &= -\arcsin \left (\frac {1}{\sqrt {1-\sqrt {\pi }\, {\mathrm e}^{x^{2}} \operatorname {erf}\left (x \right )-2 \,{\mathrm e}^{x^{2}} c_{1}}}\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 7.843 (sec). Leaf size: 66

DSolve[x*Cos[y[x]]^2*Sin[y[x]] - Sin[y[x]]^3 + Cos[y[x]]*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\cot ^{-1}\left (\sqrt {e^{x^2} \left (-\sqrt {\pi } \text {erf}(x)+4 c_1\right )}\right ) \\ y(x)\to \cot ^{-1}\left (\sqrt {e^{x^2} \left (-\sqrt {\pi } \text {erf}(x)+4 c_1\right )}\right ) \\ y(x)\to 0 \\ \end{align*}