12.5.49 problem 48

Internal problem ID [1673]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 48
Date solved : Saturday, March 29, 2025 at 11:25:36 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 102

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

\[ y = -\frac {\left (c_1 \sin \left (\frac {\ln \left (\sin \left (x \right )+1\right )}{2}+\frac {\ln \left (\sin \left (x \right )-1\right )}{2}-\ln \left (\sin \left (x \right )\right )\right )-\cos \left (\frac {\ln \left (\sin \left (x \right )+1\right )}{2}+\frac {\ln \left (\sin \left (x \right )-1\right )}{2}-\ln \left (\sin \left (x \right )\right )\right )\right ) \tan \left (x \right )}{c_1 \cos \left (\frac {\ln \left (\sin \left (x \right )+1\right )}{2}+\frac {\ln \left (\sin \left (x \right )-1\right )}{2}-\ln \left (\sin \left (x \right )\right )\right )+\sin \left (\frac {\ln \left (\sin \left (x \right )+1\right )}{2}+\frac {\ln \left (\sin \left (x \right )-1\right )}{2}-\ln \left (\sin \left (x \right )\right )\right )} \]

✓ Mathematica. Time used: 0.617 (sec). Leaf size: 20

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

\[ y(x)\to \tan (x) \tan (\log (\sin (x))-\log (\cos (x))+c_1) \]

✗ Sympy

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

NotImplementedError : The given ODE -(y(x)**2 + y(x)*tan(x) + tan(x)**2)/sin(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method