69.1.65 problem 92

Internal problem ID [14218]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 92
Date solved : Tuesday, January 28, 2025 at 06:22:23 AM
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: 0.145 (sec). Leaf size: 69

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

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

✓ Solution by Mathematica

Time used: 0.484 (sec). Leaf size: 126

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