82.16.2 problem Ex. 2

Internal problem ID [18816]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Problems at page 36
Problem number : Ex. 2
Date solved : Tuesday, January 28, 2025 at 12:21:23 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 69

dsolve(y(x)=y(x)*diff(y(x),x)^2+2*diff(y(x),x)*x,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: 0.434 (sec). Leaf size: 126

DSolve[y[x]==y[x]*D[y[x],x]^2+2*D[y[x],x]*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 -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*}