73.8.42 problem 13.6 (h)

Internal problem ID [15250]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 13. Higher order equations: Extending first order concepts. Additional exercises page 259
Problem number : 13.6 (h)
Date solved : Tuesday, January 28, 2025 at 07:50:54 AM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]]

\begin{align*} 2 x y^{\prime } y^{\prime \prime }&=-1+{y^{\prime }}^{2} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0\\ y^{\prime }\left (1\right )&=\sqrt {3} \end{align*}

✓ Solution by Maple

Time used: 0.280 (sec). Leaf size: 19

dsolve([2*x*diff(y(x),x)*diff(y(x),x$2)=diff(y(x),x)^2-1,y(1) = 0, D(y)(1) = 3^(1/2)],y(x), singsol=all)
 

\[ y = \frac {\left (2 x +1\right )^{{3}/{2}}}{3}-\sqrt {3} \]

✓ Solution by Mathematica

Time used: 0.088 (sec). Leaf size: 26

DSolve[{2*x*D[y[x],x]*D[y[x],{x,2}]==D[y[x],x]^2-1,{y[1]==0,Derivative[1][y][1]==Sqrt[3]}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{3} \left ((2 x+1)^{3/2}-3 \sqrt {3}\right ) \]