29.24.22 problem 684

Internal problem ID [5275]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 24
Problem number : 684
Date solved : Sunday, March 30, 2025 at 07:39:45 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 113

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

\begin{align*} y &= -\frac {\sqrt {-2-2 \sqrt {4 x^{4}+4 x^{2}+8 c_1 +1}}}{2} \\ y &= \frac {\sqrt {-2-2 \sqrt {4 x^{4}+4 x^{2}+8 c_1 +1}}}{2} \\ y &= -\frac {\sqrt {-2+2 \sqrt {4 x^{4}+4 x^{2}+8 c_1 +1}}}{2} \\ y &= \frac {\sqrt {-2+2 \sqrt {4 x^{4}+4 x^{2}+8 c_1 +1}}}{2} \\ \end{align*}

✓ Mathematica. Time used: 2.387 (sec). Leaf size: 151

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

\begin{align*} y(x)\to -\frac {\sqrt {-1-\sqrt {4 x^4+4 x^2+1+8 c_1}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-1-\sqrt {4 x^4+4 x^2+1+8 c_1}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {-1+\sqrt {4 x^4+4 x^2+1+8 c_1}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-1+\sqrt {4 x^4+4 x^2+1+8 c_1}}}{\sqrt {2}} \\ \end{align*}

✓ Sympy. Time used: 3.160 (sec). Leaf size: 122

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

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- \sqrt {C_{1} + 4 x^{4} + 4 x^{2}} - 1}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- \sqrt {C_{1} + 4 x^{4} + 4 x^{2}} - 1}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {\sqrt {C_{1} + 4 x^{4} + 4 x^{2}} - 1}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {\sqrt {C_{1} + 4 x^{4} + 4 x^{2}} - 1}}{2}\right ] \]