67.8.49 problem 13.8 (iv)

Internal problem ID [16544]
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 : Thursday, October 02, 2025 at 01:36:17 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

With initial conditions

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

✓ Maple. Time used: 0.145 (sec). Leaf size: 29

ode:=diff(diff(y(x),x),x) = -2*x*diff(y(x),x)^2; 
ic:=[y(1) = -1/4, D(y)(1) = 5]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 12.909 (sec). Leaf size: 26

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

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

✓ Sympy. Time used: 0.641 (sec). Leaf size: 80

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), x)**2 + Derivative(y(x), (x, 2)),0) 
ics = {y(1): -1/4, Subs(Derivative(y(x), x), x, 1): 5} 
dsolve(ode,func=y(x),ics=ics)
 

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