29.2.23 problem 48

Internal problem ID [4656]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 2
Problem number : 48
Date solved : Monday, January 27, 2025 at 09:29:39 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

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

dsolve(diff(y(x),x) = cos(x)-(sin(x)-y(x))*y(x),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: 60.943 (sec). Leaf size: 90

DSolve[D[y[x],x]==Cos[x]-(Sin[x]-y[x])*y[x],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)-\frac {e^{-\cos (x)}}{\int _1^xe^{-\cos (K[1])}dK[1]} \\ \end{align*}