73.8.53 problem 13.9 (iv)

Internal problem ID [15261]
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 : 13.9 (iv)
Date solved : Tuesday, January 28, 2025 at 07:51:06 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime }&=2 y^{\prime } y \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.132 (sec). Leaf size: 8

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

\[ y = -\tanh \left (x \right ) \]

✓ Solution by Mathematica

Time used: 1.687 (sec). Leaf size: 9

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

\[ y(x)\to -\tanh (x) \]