7.5.25 problem 25

Internal problem ID [129]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 1. First order differential equations. Section 1.6 (substitution and exact equations). Problems at page 72
Problem number : 25
Date solved : Saturday, March 29, 2025 at 04:34:48 PM
CAS classification : [_Bernoulli]

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

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

ode:=y(x)^2*(x*diff(y(x),x)+y(x))*(x^4+1)^(1/2) = 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: 4.206 (sec). Leaf size: 106

ode=y[x]^2*(x*D[y[x],x]+y[x])*(1+x^4)^(1/2)==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: 1.929 (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)
 

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