53.4.23 problem 25

Internal problem ID [8511]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 25
Date solved : Wednesday, March 05, 2025 at 06:01:54 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 24

ode:=diff(diff(y(x),x),x) = exp(x)*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_{2} c_{1} -\ln \left ({\mathrm e}^{x}-c_{1} \right )+\ln \left ({\mathrm e}^{x}\right )}{c_{1}} \]

✓ Mathematica. Time used: 0.929 (sec). Leaf size: 37

ode=D[y[x],{x,2}]==Exp[x](D[y[x],x])^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {-x+\log \left (e^x+c_1\right )+c_1 c_2}{c_1} \\ y(x)\to \text {Indeterminate} \\ y(x)\to c_2 \\ \end{align*}

✓ Sympy. Time used: 0.665 (sec). Leaf size: 31

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

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