Internal problem ID [7777]
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]
Solve \begin {gather*} \boxed {y^{\prime } \cos \relax (x )-y^{4}-y \sin \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.065 (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 \relax (x ) = \frac {\left (\cos \relax (x ) \left (c_{1} \left (\sin ^{4}\relax (x )\right )+2 \cos \relax (x ) \left (\sin ^{3}\relax (x )\right )-2 \left (\sin ^{2}\relax (x )\right ) c_{1}-3 \sin \relax (x ) \cos \relax (x )+c_{1}\right )^{2}\right )^{\frac {1}{3}}}{c_{1} \left (\sin ^{4}\relax (x )\right )+2 \cos \relax (x ) \left (\sin ^{3}\relax (x )\right )-2 \left (\sin ^{2}\relax (x )\right ) c_{1}-3 \sin \relax (x ) \cos \relax (x )+c_{1}} \\ y \relax (x ) = -\frac {\left (\cos \relax (x ) \left (c_{1} \left (\sin ^{4}\relax (x )\right )+2 \cos \relax (x ) \left (\sin ^{3}\relax (x )\right )-2 \left (\sin ^{2}\relax (x )\right ) c_{1}-3 \sin \relax (x ) \cos \relax (x )+c_{1}\right )^{2}\right )^{\frac {1}{3}}}{2 \left (c_{1} \left (\sin ^{4}\relax (x )\right )+2 \cos \relax (x ) \left (\sin ^{3}\relax (x )\right )-2 \left (\sin ^{2}\relax (x )\right ) c_{1}-3 \sin \relax (x ) \cos \relax (x )+c_{1}\right )}-\frac {i \sqrt {3}\, \left (\cos \relax (x ) \left (c_{1} \left (\sin ^{4}\relax (x )\right )+2 \cos \relax (x ) \left (\sin ^{3}\relax (x )\right )-2 \left (\sin ^{2}\relax (x )\right ) c_{1}-3 \sin \relax (x ) \cos \relax (x )+c_{1}\right )^{2}\right )^{\frac {1}{3}}}{2 \left (c_{1} \left (\sin ^{4}\relax (x )\right )+2 \cos \relax (x ) \left (\sin ^{3}\relax (x )\right )-2 \left (\sin ^{2}\relax (x )\right ) c_{1}-3 \sin \relax (x ) \cos \relax (x )+c_{1}\right )} \\ y \relax (x ) = -\frac {\left (\cos \relax (x ) \left (c_{1} \left (\sin ^{4}\relax (x )\right )+2 \cos \relax (x ) \left (\sin ^{3}\relax (x )\right )-2 \left (\sin ^{2}\relax (x )\right ) c_{1}-3 \sin \relax (x ) \cos \relax (x )+c_{1}\right )^{2}\right )^{\frac {1}{3}}}{2 \left (c_{1} \left (\sin ^{4}\relax (x )\right )+2 \cos \relax (x ) \left (\sin ^{3}\relax (x )\right )-2 \left (\sin ^{2}\relax (x )\right ) c_{1}-3 \sin \relax (x ) \cos \relax (x )+c_{1}\right )}+\frac {i \sqrt {3}\, \left (\cos \relax (x ) \left (c_{1} \left (\sin ^{4}\relax (x )\right )+2 \cos \relax (x ) \left (\sin ^{3}\relax (x )\right )-2 \left (\sin ^{2}\relax (x )\right ) c_{1}-3 \sin \relax (x ) \cos \relax (x )+c_{1}\right )^{2}\right )^{\frac {1}{3}}}{2 c_{1} \left (\sin ^{4}\relax (x )\right )+4 \cos \relax (x ) \left (\sin ^{3}\relax (x )\right )-4 \left (\sin ^{2}\relax (x )\right ) c_{1}-6 \sin \relax (x ) \cos \relax (x )+2 c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.499 (sec). Leaf size: 97
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 (x) (\cos (2 x)+2)+c_1 \cos ^3(x)}} \\ y(x)\to -\frac {\sqrt [3]{-1}}{\sqrt [3]{-\sin (x) (\cos (2 x)+2)+c_1 \cos ^3(x)}} \\ y(x)\to \frac {(-1)^{2/3}}{\sqrt [3]{-\sin (x) (\cos (2 x)+2)+c_1 \cos ^3(x)}} \\ y(x)\to 0 \\ \end{align*}