53.3.1 problem 3

Internal problem ID [8463]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 99. Clairaut equation. EXERCISES Page 320
Problem number : 3
Date solved : Monday, January 27, 2025 at 04:04:17 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} {y^{\prime }}^{2}+x^{3} y^{\prime }-2 x^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.210 (sec). Leaf size: 21

dsolve(diff(y(x),x)^2+x^3*diff(y(x),x)-2*x^2*y(x)=0,y(x), singsol=all)
 

\begin{align*} y &= -\frac {x^{4}}{8} \\ y &= c_{1} \left (x^{2}+2 c_{1} \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.510 (sec). Leaf size: 169

DSolve[(D[y[x],x])^2+x^3*D[y[x],x]-2*x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [\frac {1}{4} \log (y(x))-\frac {\sqrt {x^6+8 x^2 y(x)} \text {arctanh}\left (\frac {x^2}{\sqrt {x^4+8 y(x)}+2 \sqrt {2} \sqrt {y(x)}}\right )}{x \sqrt {x^4+8 y(x)}}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\sqrt {x^6+8 x^2 y(x)} \text {arctanh}\left (\frac {x^2}{\sqrt {x^4+8 y(x)}+2 \sqrt {2} \sqrt {y(x)}}\right )}{x \sqrt {x^4+8 y(x)}}+\frac {1}{4} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to -\frac {x^4}{8} \\ \end{align*}