60.1.469 problem 472

Internal problem ID [10483]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 472
Date solved : Monday, January 27, 2025 at 07:55:08 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} \left (x +y\right ) {y^{\prime }}^{2}+2 x y^{\prime }-y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.131 (sec). Leaf size: 119

dsolve((x+y(x))*diff(y(x),x)^2+2*x*diff(y(x),x)-y(x) = 0,y(x), singsol=all)
 

\begin{align*} y &= -\frac {\left (1+i \sqrt {3}\right ) x}{2} \\ y &= \frac {\left (i \sqrt {3}-1\right ) x}{2} \\ \ln \left (x \right )-\operatorname {arctanh}\left (\frac {2 x +y}{2 x \sqrt {\frac {y^{2}+x y+x^{2}}{x^{2}}}}\right )+\ln \left (\frac {y}{x}\right )-c_{1} &= 0 \\ \ln \left (x \right )+\operatorname {arctanh}\left (\frac {2 x +y}{2 x \sqrt {\frac {y^{2}+x y+x^{2}}{x^{2}}}}\right )+\ln \left (\frac {y}{x}\right )-c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 3.848 (sec). Leaf size: 166

DSolve[-y[x] + 2*x*D[y[x],x] + (x + y[x])*D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {2}{3} \sqrt {e^{c_1} \left (-3 x+e^{c_1}\right )}-\frac {e^{c_1}}{3} \\ y(x)\to \frac {2}{3} \sqrt {e^{c_1} \left (-3 x+e^{c_1}\right )}-\frac {e^{c_1}}{3} \\ y(x)\to e^{c_1}-2 \sqrt {e^{c_1} \left (x+e^{c_1}\right )} \\ y(x)\to 2 \sqrt {e^{c_1} \left (x+e^{c_1}\right )}+e^{c_1} \\ y(x)\to 0 \\ y(x)\to -\frac {1}{2} i \left (\sqrt {3}-i\right ) x \\ y(x)\to \frac {1}{2} i \left (\sqrt {3}+i\right ) x \\ \end{align*}