Internal problem ID [8417]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 80.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [`y=_G(x,y')`]
\[ \boxed {y^{\prime }+f \left (x \right ) \sin \left (y\right )+\left (1-f^{\prime }\left (x \right )\right ) \cos \left (y\right )=f^{\prime }\left (x \right )+1} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 41
dsolve(diff(y(x),x) + f(x)*sin(y(x)) + (1-diff(f(x),x))*cos(y(x)) - diff(f(x),x) - 1=0,y(x), singsol=all)
\[ y \left (x \right ) = -2 \arctan \left (\frac {{\mathrm e}^{\int f \left (x \right )d x}-\left (\int {\mathrm e}^{\int f \left (x \right )d x}d x \right ) f \left (x \right )-c_{1} f \left (x \right )}{c_{1} +\int {\mathrm e}^{\int f \left (x \right )d x}d x}\right ) \]
✓ Solution by Mathematica
Time used: 7.14 (sec). Leaf size: 68
DSolve[y'[x] + f[x]*Sin[y[x]] + (1-f'[x])*Cos[y[x]] - f'[x]- 1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2 \arctan \left (f(x)+\frac {\exp \left (-\int _1^x-f(K[1])dK[1]\right )}{\int _1^x-\exp \left (-\int _1^{K[2]}-f(K[1])dK[1]\right )dK[2]+c_1}\right ) \\ y(x)\to 2 \arctan (f(x)) \\ \end{align*}