Internal problem ID [6066]
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’s equation. EXERCISES Page
320
Problem number: 28.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+3 x y^{\prime }-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.17 (sec). Leaf size: 85
dsolve(diff(y(x),x)^2+3*x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\begin{align*} \frac {c_{1}}{\left (-6 x -2 \sqrt {9 x^{2}+4 y \relax (x )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}-\frac {\sqrt {9 x^{2}+4 y \relax (x )}}{5} = 0 \\ \frac {c_{1}}{\left (-6 x +2 \sqrt {9 x^{2}+4 y \relax (x )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}+\frac {\sqrt {9 x^{2}+4 y \relax (x )}}{5} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 3.234 (sec). Leaf size: 696
DSolve[(y'[x])^2+3*x*y'[x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {Root}\left [16 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+25 \text {$\#$1}^3 x^4+160 \text {$\#$1}^2 c_1{}^5 x+360 \text {$\#$1} c_1{}^5 x^3+216 c_1{}^5 x^5-64 c_1{}^{10}\&,1\right ] \\ y(x)\to \text {Root}\left [16 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+25 \text {$\#$1}^3 x^4+160 \text {$\#$1}^2 c_1{}^5 x+360 \text {$\#$1} c_1{}^5 x^3+216 c_1{}^5 x^5-64 c_1{}^{10}\&,2\right ] \\ y(x)\to \text {Root}\left [16 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+25 \text {$\#$1}^3 x^4+160 \text {$\#$1}^2 c_1{}^5 x+360 \text {$\#$1} c_1{}^5 x^3+216 c_1{}^5 x^5-64 c_1{}^{10}\&,3\right ] \\ y(x)\to \text {Root}\left [16 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+25 \text {$\#$1}^3 x^4+160 \text {$\#$1}^2 c_1{}^5 x+360 \text {$\#$1} c_1{}^5 x^3+216 c_1{}^5 x^5-64 c_1{}^{10}\&,4\right ] \\ y(x)\to \text {Root}\left [16 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+25 \text {$\#$1}^3 x^4+160 \text {$\#$1}^2 c_1{}^5 x+360 \text {$\#$1} c_1{}^5 x^3+216 c_1{}^5 x^5-64 c_1{}^{10}\&,5\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^5+2560 \text {$\#$1}^4 x^2+1600 \text {$\#$1}^3 x^4-160 \text {$\#$1}^2 c_1{}^5 x-360 \text {$\#$1} c_1{}^5 x^3-216 c_1{}^5 x^5-c_1{}^{10}\&,1\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^5+2560 \text {$\#$1}^4 x^2+1600 \text {$\#$1}^3 x^4-160 \text {$\#$1}^2 c_1{}^5 x-360 \text {$\#$1} c_1{}^5 x^3-216 c_1{}^5 x^5-c_1{}^{10}\&,2\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^5+2560 \text {$\#$1}^4 x^2+1600 \text {$\#$1}^3 x^4-160 \text {$\#$1}^2 c_1{}^5 x-360 \text {$\#$1} c_1{}^5 x^3-216 c_1{}^5 x^5-c_1{}^{10}\&,3\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^5+2560 \text {$\#$1}^4 x^2+1600 \text {$\#$1}^3 x^4-160 \text {$\#$1}^2 c_1{}^5 x-360 \text {$\#$1} c_1{}^5 x^3-216 c_1{}^5 x^5-c_1{}^{10}\&,4\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^5+2560 \text {$\#$1}^4 x^2+1600 \text {$\#$1}^3 x^4-160 \text {$\#$1}^2 c_1{}^5 x-360 \text {$\#$1} c_1{}^5 x^3-216 c_1{}^5 x^5-c_1{}^{10}\&,5\right ] \\ y(x)\to 0 \\ \end{align*}