85.27.8 problem 8

Internal problem ID [22600]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 60
Problem number : 8
Date solved : Thursday, October 02, 2025 at 08:54:51 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*}

With initial conditions

\begin{align*} y \left (0\right )&=5 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 15

ode:=diff(diff(y(x),x),x)*diff(y(x),x) = 1; 
ic:=[y(0) = 5, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\left (1+2 x \right )^{{3}/{2}}}{3}+\frac {14}{3} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 20

ode=D[y[x],{x,2}]*D[y[x],{x,1}]==1; 
ic={y[0]==5,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{3} \left ((2 x+1)^{3/2}+14\right ) \end{align*}

✓ Sympy. Time used: 0.481 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = \frac {\left (2 x + 1\right )^{\frac {3}{2}}}{3} + \frac {14}{3} \]