20.4.12 problem Problem 20

Internal problem ID [3647]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.8, Change of Variables. page 79
Problem number : Problem 20
Date solved : Monday, January 27, 2025 at 07:49:53 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.064 (sec). Leaf size: 27

dsolve(y(x)*diff(y(x),x)=sqrt(x^2+y(x)^2)-x,y(x), singsol=all)
 

\[ \frac {-c_{1} y \left (x \right )^{2}+\sqrt {x^{2}+y \left (x \right )^{2}}+x}{y \left (x \right )^{2}} = 0 \]

✓ Solution by Mathematica

Time used: 0.378 (sec). Leaf size: 57

DSolve[y[x]*D[y[x],x]==Sqrt[x^2+y[x]^2]-x,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -e^{\frac {c_1}{2}} \sqrt {2 x+e^{c_1}} \\ y(x)\to e^{\frac {c_1}{2}} \sqrt {2 x+e^{c_1}} \\ y(x)\to 0 \\ \end{align*}