8.5.25 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 : Wednesday, February 05, 2025 at 03:58:06 AM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 95

dsolve((x^4+1)^(1/2)*y(x)^2*(y(x)+x*diff(y(x),x)) = 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: 4.127 (sec). Leaf size: 106

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