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 : Friday, February 07, 2025 at 07:55:45 AM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 95

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

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

✓ Solution by Mathematica

Time used: 4.206 (sec). Leaf size: 106

DSolve[y[x]^2*(x*D[y[x],x]+y[x])*(1+x^4)^(1/2)==x,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*}