58.2.10 problem 11

Internal problem ID [9133]
Book : Second order enumerated odes
Section : section 2
Problem number : 11
Date solved : Wednesday, March 05, 2025 at 07:33:47 AM
CAS classification : [[_2nd_order, _missing_y], _Liouville, [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+{y^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 14

ode:=diff(diff(y(x),x),x)+sin(x)*diff(y(x),x)+diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \ln \left (c_{1} \left (\int {\mathrm e}^{\cos \left (x \right )}d x \right )+c_{2} \right ) \]

✓ Mathematica. Time used: 3.314 (sec). Leaf size: 63

ode=D[y[x],{x,2}]+Sin[x]*D[y[x],x]+(D[y[x],x])^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \int _1^x\frac {\exp \left (\int _1^{K[3]}-\sin (K[1])dK[1]\right )}{c_1-\int _1^{K[3]}-\exp \left (\int _1^{K[2]}-\sin (K[1])dK[1]\right )dK[2]}dK[3]+c_2 \]

✓ Sympy. Time used: 143.730 (sec). Leaf size: 37

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

\[ \left [ y{\left (x \right )} = C_{1} + \int \frac {e^{\cos {\left (x \right )}}}{C_{2} + \int e^{\cos {\left (x \right )}}\, dx}\, dx, \ y{\left (x \right )} = C_{1} + \int \frac {e^{\cos {\left (x \right )}}}{C_{2} + \int e^{\cos {\left (x \right )}}\, dx}\, dx\right ] \]