75.14.36 problem 362

Internal problem ID [16879]
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 : 362
Date solved : Monday, March 31, 2025 at 03:34:32 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.053 (sec). Leaf size: 22

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

\begin{align*} y &= 0 \\ y &= \frac {4}{\left (c_1 \sin \left (x \right )-c_2 \cos \left (x \right )\right )^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.601 (sec). Leaf size: 44

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

\[ y(x)\to c_2 \exp \left (\int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+4}dK[1]\&\right ]\left [c_1+\frac {K[2]}{2}\right ]dK[2]\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(6)*sqrt((-2*y(x) + Derivative(y(x), (x, 2)))*y(x))/3 + Derivative(y(x), x) cannot be solved by the factorable group method