73.8.49 problem 13.8 (iv)

Internal problem ID [15257]
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 : 13.8 (iv)
Date solved : Tuesday, January 28, 2025 at 07:51:01 AM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

With initial conditions

\begin{align*} y \left (1\right )&=-{\frac {1}{4}}\\ y^{\prime }\left (1\right )&=5 \end{align*}

✓ Solution by Maple

Time used: 0.098 (sec). Leaf size: 29

dsolve([diff(y(x),x$2)=-2*x*diff(y(x),x)^2,y(1) = -1/4, D(y)(1) = 5],y(x), singsol=all)
 

\[ y = -\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {5}\, x}{2}\right )}{2}+\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {5}}{2}\right )}{2}-\frac {1}{4} \]

✓ Solution by Mathematica

Time used: 30.116 (sec). Leaf size: 26

DSolve[{D[y[x],{x,2}]==-2*x*D[y[x],x]^2,{y[1]==-1/4,Derivative[1][y][1]==5}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \int _1^x\frac {1}{K[1]^2-\frac {4}{5}}dK[1]-\frac {1}{4} \]