8.5.12 problem 12

Internal problem ID [740]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.6, Substitution methods and exact equations. Page 74
Problem number : 12
Date solved : Wednesday, February 05, 2025 at 03:51:40 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} x y y^{\prime }&=y^{2}+x \sqrt {4 x^{2}+y^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 30

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

\[ \frac {x \ln \left (x \right )-c_1 x -\sqrt {4 x^{2}+y^{2}}}{x} = 0 \]

✓ Solution by Mathematica

Time used: 0.259 (sec). Leaf size: 54

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

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