60.1.196 problem 197

Internal problem ID [10210]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 197
Date solved : Monday, January 27, 2025 at 06:38:40 PM
CAS classification : [_Bernoulli]

\begin{align*} \cos \left (x \right ) y^{\prime }-y^{4}-y \sin \left (x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 179

dsolve(cos(x)*diff(y(x),x) - y(x)^4 - y(x)*sin(x)=0,y(x), singsol=all)
 

\begin{align*} y &= \frac {\sec \left (x \right ) \left (\cos \left (x \right )^{3} \left (-c_{1} \cos \left (x \right )^{3}+2 \cos \left (x \right )^{2} \sin \left (x \right )+\sin \left (x \right )\right )^{2}\right )^{{1}/{3}}}{c_{1} \cos \left (x \right )^{3}-2 \cos \left (x \right )^{2} \sin \left (x \right )-\sin \left (x \right )} \\ y &= \frac {\sec \left (x \right ) \left (\cos \left (x \right )^{3} \left (-c_{1} \cos \left (x \right )^{3}+2 \cos \left (x \right )^{2} \sin \left (x \right )+\sin \left (x \right )\right )^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{-2 c_{1} \cos \left (x \right )^{3}+4 \cos \left (x \right )^{2} \sin \left (x \right )+2 \sin \left (x \right )} \\ y &= \frac {\left (i \sqrt {3}-1\right ) \sec \left (x \right ) \left (\cos \left (x \right )^{3} \left (-c_{1} \cos \left (x \right )^{3}+2 \cos \left (x \right )^{2} \sin \left (x \right )+\sin \left (x \right )\right )^{2}\right )^{{1}/{3}}}{2 c_{1} \cos \left (x \right )^{3}-4 \cos \left (x \right )^{2} \sin \left (x \right )-2 \sin \left (x \right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.104 (sec). Leaf size: 109

DSolve[Cos[x]*D[y[x],x] - y[x]^4 - y[x]*Sin[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{\sqrt [3]{-\sin ^3(x)+c_1 \cos ^3(x)-3 \sin (x) \cos ^2(x)}} \\ y(x)\to -\frac {\sqrt [3]{-1}}{\sqrt [3]{-\sin ^3(x)+c_1 \cos ^3(x)-3 \sin (x) \cos ^2(x)}} \\ y(x)\to \frac {(-1)^{2/3}}{\sqrt [3]{-\sin ^3(x)+c_1 \cos ^3(x)-3 \sin (x) \cos ^2(x)}} \\ y(x)\to 0 \\ \end{align*}