67.8.46 problem 13.8 (i)

Internal problem ID [16541]
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 (i)
Date solved : Thursday, October 02, 2025 at 01:36:15 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 (0\right )&=3 \\ y^{\prime }\left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.096 (sec). Leaf size: 12

ode:=diff(diff(y(x),x),x) = -2*x*diff(y(x),x)^2; 
ic:=[y(0) = 3, D(y)(0) = 4]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = 2 \arctan \left (2 x \right )+3 \]

✓ Mathematica. Time used: 14.107 (sec). Leaf size: 43

ode=D[y[x],{x,2}]==-2*x*D[y[x],x]^2; 
ic={y[0]==3,Derivative[1][y][0] ==4}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x\frac {1}{K[1]^2+\frac {1}{4}}dK[1]-\int _1^0\frac {1}{K[1]^2+\frac {1}{4}}dK[1]+3 \end{align*}

✓ Sympy. Time used: 0.604 (sec). Leaf size: 22

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(0): 3, Subs(Derivative(y(x), x), x, 0): 4} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - i \log {\left (x - \frac {i}{2} \right )} + i \log {\left (x + \frac {i}{2} \right )} + 3 + \pi \]