53.4.30 problem 33

Internal problem ID [8518]
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
Date solved : Monday, January 27, 2025 at 04:08:55 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} \left (y y^{\prime \prime }+1+{y^{\prime }}^{2}\right )^{2}&=\left (1+{y^{\prime }}^{2}\right )^{3} \end{align*}

✓ Solution by Maple

Time used: 0.144 (sec). Leaf size: 105

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

✓ Solution by Mathematica

Time used: 48.494 (sec). Leaf size: 194

DSolve[(y[x]*D[y[x],{x,2}]+1+(D[y[x],x])^2)^2==(1+(D[y[x],x])^2)^3,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\left (\left (\cosh \left (\frac {c_1}{2}\right )+\sinh \left (\frac {c_1}{2}\right )\right ) \left (\sqrt {\left (x^2+2 c_2 x+1+c_2{}^2\right ) \sinh (c_1)-\left (x^2+2 c_2 x-1+c_2{}^2\right ) \cosh (c_1)}+\cosh \left (\frac {c_1}{2}\right )+\sinh \left (\frac {c_1}{2}\right )\right )\right ) \\ y(x)\to \left (\cosh \left (\frac {c_1}{2}\right )+\sinh \left (\frac {c_1}{2}\right )\right ) \left (\sqrt {\left (x^2+2 c_2 x+1+c_2{}^2\right ) \sinh (c_1)-\left (x^2+2 c_2 x-1+c_2{}^2\right ) \cosh (c_1)}-\cosh \left (\frac {c_1}{2}\right )-\sinh \left (\frac {c_1}{2}\right )\right ) \\ y(x)\to -\sqrt {-(x+c_2){}^2} \\ y(x)\to \sqrt {-(x+c_2){}^2} \\ \end{align*}