60.1.21 problem 21

Internal problem ID [10035]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 21
Date solved : Monday, January 27, 2025 at 06:18:58 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }-y^{2}+y \sin \left (x \right )-\cos \left (x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 38

dsolve(diff(y(x),x) - y(x)^2 +y(x)*sin(x) - cos(x)=0,y(x), singsol=all)
 

\[ y = \frac {\sin \left (x \right ) \left (\int {\mathrm e}^{-\cos \left (x \right )}d x \right )+\sin \left (x \right ) c_{1} -{\mathrm e}^{-\cos \left (x \right )}}{c_{1} +\int {\mathrm e}^{-\cos \left (x \right )}d x} \]

✓ Solution by Mathematica

Time used: 0.999 (sec). Leaf size: 140

DSolve[D[y[x],x] - y[x]^2 +y[x]*Sin[x] - Cos[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {c_1 \left (-\exp \left (-\int _1^x-\sin (K[1])dK[1]\right )\right )+c_1 \sin (x) \int _1^x\exp \left (-\int _1^{K[2]}-\sin (K[1])dK[1]\right )dK[2]+\sin (x)}{1+c_1 \int _1^x\exp \left (-\int _1^{K[2]}-\sin (K[1])dK[1]\right )dK[2]} \\ y(x)\to \sin (x)-\frac {\exp \left (-\int _1^x-\sin (K[1])dK[1]\right )}{\int _1^x\exp \left (-\int _1^{K[2]}-\sin (K[1])dK[1]\right )dK[2]} \\ \end{align*}