Internal problem ID [8358]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 21.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}+y \sin \left (x \right )=\cos \left (x \right )} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 38
dsolve(diff(y(x),x) - y(x)^2 +y(x)*sin(x) - cos(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sin \left (x \right ) \left (\int {\mathrm e}^{-\cos \left (x \right )}d x \right )+c_{1} \sin \left (x \right )-{\mathrm e}^{-\cos \left (x \right )}}{c_{1} +\int {\mathrm e}^{-\cos \left (x \right )}d x} \]
✓ Solution by Mathematica
Time used: 42.767 (sec). Leaf size: 158
DSolve[y'[x] - y[x]^2 +y[x]*Sin[x] - Cos[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 \sin (x) \int _1^xe^{-\cos (K[1])}dK[1]+\sin (x)+c_1 \left (-e^{-\cos (x)}\right )}{1+c_1 \int _1^xe^{-\cos (K[1])}dK[1]} \\ y(x)\to \sin (x) \\ y(x)\to \frac {\sin ^3(x) e^{\cos (x)} \int _1^{\cos (x)}\frac {e^{-K[1]} K[1]}{\left (1-K[1]^2\right )^{3/2}}dK[1]}{\sin ^2(x) e^{\cos (x)} \int _1^{\cos (x)}\frac {e^{-K[1]} K[1]}{\left (1-K[1]^2\right )^{3/2}}dK[1]-\sqrt {\sin ^2(x)}} \\ \end{align*}