62.9.6 problem Ex 6

Internal problem ID [12766]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 16. Integrating factors by inspection. Page 23
Problem number : Ex 6
Date solved : Monday, March 31, 2025 at 07:03:35 AM
CAS classification : [[_homogeneous, `class D`], _rational, _Riccati]

\begin{align*} x y^{\prime }-y&=x^{2}+y^{2} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 10

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

\[ y = \tan \left (x +c_1 \right ) x \]

✓ Mathematica. Time used: 0.091 (sec). Leaf size: 28

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {1}{K[1]^2+1}dK[1]=x+c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.313 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = \frac {x \left (i C_{1} + i e^{2 i x}\right )}{C_{1} - e^{2 i x}} \]