67.8.27 problem 13.5 (c)

Internal problem ID [16522]
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 : 13.5 (c)
Date solved : Thursday, October 02, 2025 at 01:35:58 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]]

\begin{align*} y^{\prime } y^{\prime \prime }&=1 \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 40

ode:=diff(y(x),x)*diff(diff(y(x),x),x) = 1; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\left (2 c_1 +2 x \right )^{{3}/{2}}}{3}+c_2 \\ y &= \frac {\left (-2 c_1 -2 x \right ) \sqrt {2 c_1 +2 x}}{3}+c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 49

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

\begin{align*} y(x)&\to c_2-\frac {2}{3} \sqrt {2} (x+c_1){}^{3/2}\\ y(x)&\to \frac {2}{3} \sqrt {2} (x+c_1){}^{3/2}+c_2 \end{align*}

✓ Sympy. Time used: 0.456 (sec). Leaf size: 27

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

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