Internal problem ID [7776]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 197.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
\[ \boxed {y^{\prime } \cos \left (x \right )-y^{4}-y \sin \left (x \right )=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 364
dsolve(cos(x)*diff(y(x),x) - y(x)^4 - y(x)*sin(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {{\left (\cos \left (x \right ) \left (c_{1} \sin \left (x \right )^{4}+2 \cos \left (x \right ) \sin \left (x \right )^{3}-2 \sin \left (x \right )^{2} c_{1} -3 \cos \left (x \right ) \sin \left (x \right )+c_{1} \right )^{2}\right )}^{\frac {1}{3}}}{c_{1} \sin \left (x \right )^{4}+2 \cos \left (x \right ) \sin \left (x \right )^{3}-2 \sin \left (x \right )^{2} c_{1} -3 \cos \left (x \right ) \sin \left (x \right )+c_{1}} \\ y \left (x \right ) = -\frac {{\left (\cos \left (x \right ) \left (c_{1} \sin \left (x \right )^{4}+2 \cos \left (x \right ) \sin \left (x \right )^{3}-2 \sin \left (x \right )^{2} c_{1} -3 \cos \left (x \right ) \sin \left (x \right )+c_{1} \right )^{2}\right )}^{\frac {1}{3}}}{2 \left (c_{1} \sin \left (x \right )^{4}+2 \cos \left (x \right ) \sin \left (x \right )^{3}-2 \sin \left (x \right )^{2} c_{1} -3 \cos \left (x \right ) \sin \left (x \right )+c_{1} \right )}-\frac {i \sqrt {3}\, {\left (\cos \left (x \right ) \left (c_{1} \sin \left (x \right )^{4}+2 \cos \left (x \right ) \sin \left (x \right )^{3}-2 \sin \left (x \right )^{2} c_{1} -3 \cos \left (x \right ) \sin \left (x \right )+c_{1} \right )^{2}\right )}^{\frac {1}{3}}}{2 \left (c_{1} \sin \left (x \right )^{4}+2 \cos \left (x \right ) \sin \left (x \right )^{3}-2 \sin \left (x \right )^{2} c_{1} -3 \cos \left (x \right ) \sin \left (x \right )+c_{1} \right )} \\ y \left (x \right ) = -\frac {{\left (\cos \left (x \right ) \left (c_{1} \sin \left (x \right )^{4}+2 \cos \left (x \right ) \sin \left (x \right )^{3}-2 \sin \left (x \right )^{2} c_{1} -3 \cos \left (x \right ) \sin \left (x \right )+c_{1} \right )^{2}\right )}^{\frac {1}{3}}}{2 \left (c_{1} \sin \left (x \right )^{4}+2 \cos \left (x \right ) \sin \left (x \right )^{3}-2 \sin \left (x \right )^{2} c_{1} -3 \cos \left (x \right ) \sin \left (x \right )+c_{1} \right )}+\frac {i \sqrt {3}\, {\left (\cos \left (x \right ) \left (c_{1} \sin \left (x \right )^{4}+2 \cos \left (x \right ) \sin \left (x \right )^{3}-2 \sin \left (x \right )^{2} c_{1} -3 \cos \left (x \right ) \sin \left (x \right )+c_{1} \right )^{2}\right )}^{\frac {1}{3}}}{2 c_{1} \sin \left (x \right )^{4}+4 \cos \left (x \right ) \sin \left (x \right )^{3}-4 \sin \left (x \right )^{2} c_{1} -6 \cos \left (x \right ) \sin \left (x \right )+2 c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.833 (sec). Leaf size: 109
DSolve[Cos[x]*y'[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*}