32.6.22 problem Exercise 12.22, page 103

Internal problem ID [5887]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number : Exercise 12.22, page 103
Date solved : Sunday, March 30, 2025 at 10:23:49 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 43

ode:=(y(x)^2+a*sin(x))*diff(y(x),x) = cos(x); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {\left (-a^{3} \sin \left (x \right )-a^{2} y^{2}-2 a y-2\right ) {\mathrm e}^{-a y}+c_1 \,a^{3}}{a^{3}} = 0 \]

✓ Mathematica. Time used: 0.199 (sec). Leaf size: 45

ode=(y[x]^2+a*Sin[x])*D[y[x],x]==Cos[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\sin (x) \left (-e^{-a y(x)}\right )-\frac {e^{-a y(x)} \left (a^2 y(x)^2+2 a y(x)+2\right )}{a^3}=c_1,y(x)\right ] \]

✓ Sympy. Time used: 3.338 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq((a*sin(x) + y(x)**2)*Derivative(y(x), x) - cos(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} - \begin {cases} \frac {\left (- a^{2} y^{2}{\left (x \right )} - 2 a y{\left (x \right )} - 2\right ) e^{- a y{\left (x \right )}}}{a^{3}} & \text {for}\: a^{3} \neq 0 \\\frac {y^{3}{\left (x \right )}}{3} & \text {otherwise} \end {cases} + e^{- a y{\left (x \right )}} \sin {\left (x \right )} = 0 \]