53.4.40 problem 43

Internal problem ID [8528]
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 : 43
Date solved : Sunday, March 30, 2025 at 01:13:35 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} 4 y {y^{\prime }}^{2} y^{\prime \prime }&={y^{\prime }}^{4}+3 \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 109

ode:=4*y(x)*diff(y(x),x)^2*diff(diff(y(x),x),x) = diff(y(x),x)^4+3; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \frac {-4 \left (c_1 y-3\right )^{{3}/{4}}+\left (-3 x -3 c_2 \right ) c_1}{3 c_1} &= 0 \\ \frac {4 \left (c_1 y-3\right )^{{3}/{4}}+\left (-3 x -3 c_2 \right ) c_1}{3 c_1} &= 0 \\ \frac {-4 i \left (c_1 y-3\right )^{{3}/{4}}+\left (-3 x -3 c_2 \right ) c_1}{3 c_1} &= 0 \\ \frac {4 i \left (c_1 y-3\right )^{{3}/{4}}+\left (-3 x -3 c_2 \right ) c_1}{3 c_1} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 60.252 (sec). Leaf size: 156

ode=4*y[x]*(D[y[x],x])^2*D[y[x],{x,2}]==(D[y[x],x])^4+3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {3}{8} e^{-4 c_1} \left (8+\sqrt [3]{6} \left (-e^{4 c_1} (x+c_2)\right ){}^{4/3}\right ) \\ y(x)\to \frac {3}{8} e^{-4 c_1} \left (8+\sqrt [3]{6} \left (-i e^{4 c_1} (x+c_2)\right ){}^{4/3}\right ) \\ y(x)\to \frac {3}{8} e^{-4 c_1} \left (8+\sqrt [3]{6} \left (i e^{4 c_1} (x+c_2)\right ){}^{4/3}\right ) \\ y(x)\to \frac {3}{8} e^{-4 c_1} \left (8+\sqrt [3]{6} \left (e^{4 c_1} (x+c_2)\right ){}^{4/3}\right ) \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x)*Derivative(y(x), x)**2*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**4 - 3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(sqrt(4*y(x)**2*Derivative(y(x), (x, 2))**2 - 3) + 2*y(x)*Derivative(y(x), (x, 2))) + Derivative(y(x), x) cannot be solved by the factorable group method