15.2.16 problem 16

Internal problem ID [2886]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 6, page 25
Problem number : 16
Date solved : Monday, January 27, 2025 at 06:24:50 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} \left (\frac {x}{y}+\frac {y}{x}\right ) y^{\prime }+1&=0 \end{align*}

✓ Solution by Maple

Time used: 0.494 (sec). Leaf size: 221

dsolve((x/y(x)+y(x)/x)*diff(y(x),x)+1=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.102 (sec). Leaf size: 121

DSolve[(x/y[x]+y[x]/x)*D[y[x],x]+1==0,y[x],x,IncludeSingularSolutions -> True]
 

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