Internal problem ID [6850]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing.
EXERCISES Page 324
Problem number: 33.
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {\left (y y^{\prime \prime }+1+{y^{\prime }}^{2}\right )^{2}-\left (1+{y^{\prime }}^{2}\right )^{3}=0} \]
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 107
dsolve((y(x)*diff(y(x),x$2)+1+diff(y(x),x)^2)^2=(1+diff(y(x),x)^2)^3,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= -c_{1} -\sqrt {-\left (x +c_{1} +c_{2} \right ) \left (x -c_{1} +c_{2} \right )} \\ y \left (x \right ) &= -c_{1} +\sqrt {-\left (x +c_{1} +c_{2} \right ) \left (x -c_{1} +c_{2} \right )} \\ y \left (x \right ) &= c_{1} -\sqrt {-\left (x +c_{1} +c_{2} \right ) \left (x -c_{1} +c_{2} \right )} \\ y \left (x \right ) &= c_{1} +\sqrt {-\left (x +c_{1} +c_{2} \right ) \left (x -c_{1} +c_{2} \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 45.659 (sec). Leaf size: 155
DSolve[(y[x]*y''[x]+1+(y'[x])^2)^2==(1+(y'[x])^2)^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {e^{2 c_1}-(x+c_2){}^2}-e^{c_1} \\ y(x)\to e^{c_1}-\sqrt {e^{2 c_1}-(x+c_2){}^2} \\ y(x)\to \sqrt {e^{2 c_1}-(x+c_2){}^2}-e^{c_1} \\ y(x)\to \sqrt {e^{2 c_1}-(x+c_2){}^2}+e^{c_1} \\ y(x)\to -\sqrt {-(x+c_2){}^2} \\ y(x)\to \sqrt {-(x+c_2){}^2} \\ \end{align*}