60.1.80 problem 80

Internal problem ID [10094]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 80
Date solved : Tuesday, January 28, 2025 at 04:25:31 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} 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&=0 \end{align*}

✓ Solution by Maple

Time used: 0.062 (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)
 

\[ y = 2 \arctan \left (\frac {-{\mathrm e}^{\int fd x}+\left (\int {\mathrm e}^{\int fd x}d x \right ) f+f c_{1}}{c_{1} +\int {\mathrm e}^{\int fd x}d x}\right ) \]

✓ Solution by Mathematica

Time used: 7.206 (sec). Leaf size: 68

DSolve[D[y[x],x] + f[x]*Sin[y[x]] + (1-D[ f[x],x])*Cos[y[x]] - D[ f[x],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*}