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 : Sunday, March 30, 2025 at 03:36:11 AM
CAS classification : [_Bernoulli]

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

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

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

\[ y = \frac {a^{2}+k^{2}}{c_1 \left (a^{2}+k^{2}\right ) {\mathrm e}^{-a x}-b \left (k \sin \left (k x \right )+\cos \left (k x \right ) a \right )} \]

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

ode=D[y[x],x]==(a+b y[x] Cos[k x])y[x]; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 1.636 (sec). Leaf size: 180

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
k = symbols("k") 
y = Function("y") 
ode = Eq((-a - b*y(x)*cos(k*x))*y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \begin {cases} \frac {1}{C_{1} - b x} & \text {for}\: a = 0 \wedge k = 0 \\\frac {2 k e^{a x + i k x}}{2 C_{1} k e^{i k x} - b k x e^{i k x} - b \sin {\left (k x \right )}} & \text {for}\: a = - i k \\\frac {2 k e^{a x}}{2 C_{1} k - b k x - b e^{i k x} \sin {\left (k x \right )}} & \text {for}\: a = i k \\\frac {a^{2} e^{a x}}{C_{1} a^{2} + C_{1} k^{2} - a b e^{a x} \cos {\left (k x \right )} - b k e^{a x} \sin {\left (k x \right )}} + \frac {k^{2} e^{a x}}{C_{1} a^{2} + C_{1} k^{2} - a b e^{a x} \cos {\left (k x \right )} - b k e^{a x} \sin {\left (k x \right )}} & \text {otherwise} \end {cases} \]