51.1.27 problem 27

Internal problem ID [10297]
Book : First order enumerated odes
Section : section 1
Problem number : 27
Date solved : Tuesday, September 30, 2025 at 07:18:25 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 59

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

\[ y = \frac {-c_1 \operatorname {MathieuSPrime}\left (0, -2, -\frac {\pi }{4}+\frac {x}{2}\right )-\operatorname {MathieuCPrime}\left (0, -2, -\frac {\pi }{4}+\frac {x}{2}\right )}{2 c_1 \operatorname {MathieuS}\left (0, -2, -\frac {\pi }{4}+\frac {x}{2}\right )+2 \operatorname {MathieuC}\left (0, -2, -\frac {\pi }{4}+\frac {x}{2}\right )} \]

✓ Mathematica. Time used: 0.107 (sec). Leaf size: 105

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

\begin{align*} y(x)&\to \frac {-\text {MathieuSPrime}\left [0,-2,\frac {1}{4} (\pi -2 x)\right ]+c_1 \text {MathieuCPrime}\left [0,-2,\frac {1}{4} (\pi -2 x)\right ]}{2 \left (\text {MathieuS}\left [0,-2,\frac {1}{4} (2 x-\pi )\right ]+c_1 \text {MathieuC}\left [0,-2,\frac {1}{4} (\pi -2 x)\right ]\right )}\\ y(x)&\to \frac {\text {MathieuCPrime}\left [0,-2,\frac {1}{4} (\pi -2 x)\right ]}{2 \text {MathieuC}\left [0,-2,\frac {1}{4} (\pi -2 x)\right ]} \end{align*}

✗ Sympy

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

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