75.14.19 problem 345

Internal problem ID [16862]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 14. Differential equations admitting of depression of their order. Exercises page 107
Problem number : 345
Date solved : Monday, March 31, 2025 at 03:33:38 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.598 (sec). Leaf size: 34

ode:=diff(diff(y(x),x),x) = (1+diff(y(x),x))^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -x +c_{1} \\ y &= \frac {1}{12} x^{3}+\frac {1}{4} c_{1} x^{2}+\frac {1}{4} c_{1}^{2} x -x +c_{2} \\ \end{align*}

✓ Mathematica. Time used: 0.067 (sec). Leaf size: 30

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

\[ y(x)\to \frac {1}{12} x \left (x^2+3 c_1 x+3 \left (-4+c_1{}^2\right )\right )+c_2 \]

✓ Sympy. Time used: 0.503 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = C_{1} + \frac {C_{2}^{2} x}{4} - \frac {C_{2} x^{2}}{4} + \frac {x^{3}}{12} - x \]