2.5.25 problem 25

Internal problem ID [753]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.6, Substitution methods and exact equations. Page 74
Problem number : 25
Date solved : Tuesday, September 30, 2025 at 04:09:13 AM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 95

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

\begin{align*} y &= \frac {\left (3 \int \frac {x^{3}}{\sqrt {x^{4}+1}}d x +c_1 \right )^{{1}/{3}}}{x} \\ y &= -\frac {\left (3 \int \frac {x^{3}}{\sqrt {x^{4}+1}}d x +c_1 \right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 x} \\ y &= \frac {\left (3 \int \frac {x^{3}}{\sqrt {x^{4}+1}}d x +c_1 \right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 x} \\ \end{align*}

✓ Mathematica. Time used: 4.015 (sec). Leaf size: 106

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

\begin{align*} y(x)&\to \sqrt [3]{\frac {3 \sqrt {x^4+1}}{2 x^3}+\frac {c_1}{x^3}}\\ y(x)&\to -\sqrt [3]{-\frac {1}{2}} \sqrt [3]{\frac {3 \sqrt {x^4+1}+2 c_1}{x^3}}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{\frac {3 \sqrt {x^4+1}}{2 x^3}+\frac {c_1}{x^3}} \end{align*}

✓ Sympy. Time used: 1.258 (sec). Leaf size: 100

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

\[ \left [ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \sqrt [3]{\frac {C_{1} + 3 \sqrt {x^{4} + 1}}{x^{3}}}}{2}, \ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \sqrt [3]{\frac {C_{1} + 3 \sqrt {x^{4} + 1}}{x^{3}}} \left (-1 - \sqrt {3} i\right )}{4}, \ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \sqrt [3]{\frac {C_{1} + 3 \sqrt {x^{4} + 1}}{x^{3}}} \left (-1 + \sqrt {3} i\right )}{4}\right ] \]