29.3.16 problem 70

Internal problem ID [4678]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 3
Problem number : 70
Date solved : Monday, January 27, 2025 at 09:31:13 AM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }&=\left (a +b y \cos \left (k x \right )\right ) y \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 45

dsolve(diff(y(x),x) = (a+b*y(x)*cos(k*x))*y(x),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {a^{2}+k^{2}}{c_{1} \left (a^{2}+k^{2}\right ) {\mathrm e}^{-a x}-b \left (k \sin \left (k x \right )+\cos \left (k x \right ) a \right )} \]

✓ Solution by Mathematica

Time used: 0.211 (sec). Leaf size: 62

DSolve[D[y[x],x]==(a+b y[x] Cos[k x])y[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\left (a^2+k^2\right ) e^{a x}}{-\left (c_1 \left (a^2+k^2\right )\right )+b k e^{a x} \sin (k x)+a b e^{a x} \cos (k x)} \\ y(x)\to 0 \\ \end{align*}