69.1.73 problem 117

Internal problem ID [14147]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 117
Date solved : Wednesday, March 05, 2025 at 10:36:47 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_poly_yn]]

\begin{align*} y^{\prime \prime }&=\frac {1}{2 y^{\prime }} \end{align*}

✓ Maple. Time used: 0.070 (sec). Leaf size: 39

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

\begin{align*} y &= \frac {\left (2 c_{1} +2 x \right ) \sqrt {x +c_{1}}}{3}+c_{2} \\ y &= \frac {\left (-2 c_{1} -2 x \right ) \sqrt {x +c_{1}}}{3}+c_{2} \\ \end{align*}

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 43

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

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

✓ Sympy. Time used: 0.678 (sec). Leaf size: 27

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

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