Internal problem ID [6819]
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]
\[ \boxed {{y^{\prime }}^{2}+3 y^{\prime } x -y=0} \]
✓ Solution by Maple
Time used: 0.078 (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 \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}-\frac {\sqrt {9 x^{2}+4 y \left (x \right )}}{5} = 0 \frac {c_{1}}{\left (-6 x +2 \sqrt {9 x^{2}+4 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}+\frac {\sqrt {9 x^{2}+4 y \left (x \right )}}{5} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 14.495 (sec). Leaf size: 776
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 {$\#$1}^5+40 \text {$\#$1}^4 x^2+25 \text {$\#$1}^3 x^4+160 \text {$\#$1}^2 e^{5 c_1} x+360 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-64 e^{10 c_1}\&,1\right ] y(x)\to \text {Root}\left [16 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+25 \text {$\#$1}^3 x^4+160 \text {$\#$1}^2 e^{5 c_1} x+360 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-64 e^{10 c_1}\&,2\right ] y(x)\to \text {Root}\left [16 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+25 \text {$\#$1}^3 x^4+160 \text {$\#$1}^2 e^{5 c_1} x+360 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-64 e^{10 c_1}\&,3\right ] y(x)\to \text {Root}\left [16 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+25 \text {$\#$1}^3 x^4+160 \text {$\#$1}^2 e^{5 c_1} x+360 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-64 e^{10 c_1}\&,4\right ] y(x)\to \text {Root}\left [16 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+25 \text {$\#$1}^3 x^4+160 \text {$\#$1}^2 e^{5 c_1} x+360 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-64 e^{10 c_1}\&,5\right ] y(x)\to \text {Root}\left [1024 \text {$\#$1}^5+2560 \text {$\#$1}^4 x^2+1600 \text {$\#$1}^3 x^4-160 \text {$\#$1}^2 e^{5 c_1} x-360 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-e^{10 c_1}\&,1\right ] y(x)\to \text {Root}\left [1024 \text {$\#$1}^5+2560 \text {$\#$1}^4 x^2+1600 \text {$\#$1}^3 x^4-160 \text {$\#$1}^2 e^{5 c_1} x-360 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-e^{10 c_1}\&,2\right ] y(x)\to \text {Root}\left [1024 \text {$\#$1}^5+2560 \text {$\#$1}^4 x^2+1600 \text {$\#$1}^3 x^4-160 \text {$\#$1}^2 e^{5 c_1} x-360 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-e^{10 c_1}\&,3\right ] y(x)\to \text {Root}\left [1024 \text {$\#$1}^5+2560 \text {$\#$1}^4 x^2+1600 \text {$\#$1}^3 x^4-160 \text {$\#$1}^2 e^{5 c_1} x-360 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-e^{10 c_1}\&,4\right ] y(x)\to \text {Root}\left [1024 \text {$\#$1}^5+2560 \text {$\#$1}^4 x^2+1600 \text {$\#$1}^3 x^4-160 \text {$\#$1}^2 e^{5 c_1} x-360 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-e^{10 c_1}\&,5\right ] y(x)\to 0 \end{align*}