81.1.8 problem 7

Internal problem ID [18603]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter I. Introduction. Exercises at page 13
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 12:03:52 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y-2 x y^{\prime }-y {y^{\prime }}^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.573 (sec). Leaf size: 69

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

\begin{align*} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \sqrt {c_{1} \left (c_{1} -2 x \right )} \\ y \left (x \right ) &= \sqrt {c_{1} \left (c_{1} +2 x \right )} \\ y \left (x \right ) &= -\sqrt {c_{1} \left (c_{1} -2 x \right )} \\ y \left (x \right ) &= -\sqrt {c_{1} \left (c_{1} +2 x \right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.453 (sec). Leaf size: 126

DSolve[y[x]-2*x*D[y[x],x]-y[x]*D[y[x],x]^2==0,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 -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 \\ y(x)\to -i x \\ y(x)\to i x \\ \end{align*}