75.6.32 problem 165

Internal problem ID [16788]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 6. Linear equations of the first order. The Bernoulli equation. Exercises page 54
Problem number : 165
Date solved : Tuesday, January 28, 2025 at 09:22:57 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 15

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

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

✓ Solution by Mathematica

Time used: 0.268 (sec). Leaf size: 49

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

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] (K[1]+1)}dK[1]\&\right ]\left [\int _1^x\cos (K[2])dK[2]+c_1\right ] \\ y(x)\to -1 \\ y(x)\to 0 \\ \end{align*}