58.5.37 problem 41

Internal problem ID [14631]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.3 (Linear equations). Exercises page 56
Problem number : 41
Date solved : Thursday, October 02, 2025 at 09:44:49 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime }&=-8 x y^{2}+4 x \left (1+4 x \right ) y-8 x^{3}-4 x^{2}+1 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 35

ode:=diff(y(x),x) = -8*x*y(x)^2+4*x*(1+4*x)*y(x)-8*x^3-4*x^2+1; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1 \left (1+2 x \right ) {\mathrm e}^{2 x^{2}}+2 x}{2 c_1 \,{\mathrm e}^{2 x^{2}}+2} \]

✓ Mathematica. Time used: 0.123 (sec). Leaf size: 30

ode=D[y[x],x]==-8*x*y[x]^2+4*x*(4*x+1)*y[x]-(8*x^3+4*x^2-1); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{4} \left (\tanh \left (x^2+i c_1\right )+4 x+1\right )\\ y(x)&\to \text {Indeterminate} \end{align*}

✓ Sympy. Time used: 0.226 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*x**3 + 4*x**2 - 4*x*(4*x + 1)*y(x) + 8*x*y(x)**2 + Derivative(y(x), x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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