73.8.44 problem 13.7 (d)

Internal problem ID [15173]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 13. Higher order equations: Extending first order concepts. Additional exercises page 259
Problem number : 13.7 (d)
Date solved : Thursday, March 13, 2025 at 05:48:25 AM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&={\frac {3}{4}} \end{align*}

✓ Maple. Time used: 0.223 (sec). Leaf size: 14

ode:=y(x)*diff(diff(y(x),x),x)+2*diff(y(x),x)^2 = 3*y(x)*diff(y(x),x); 
ic:=y(0) = 2, D(y)(0) = 3/4; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \left (3 \,{\mathrm e}^{3 x}+5\right )^{{1}/{3}} \]

✓ Mathematica. Time used: 0.495 (sec). Leaf size: 118

ode=y[x]*D[y[x],{x,2}]+2*D[y[x],x]^2==3*y[x]*D[y[x],x]; 
ic={y[0]==2,Derivative[1][y][0] ==3/4}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to 2 \exp \left (\int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) K[1]}dK[1]\&\right ]\left [\int _1^{\frac {3}{8}}\frac {1}{(K[1]-1) K[1]}dK[1]-3 K[2]\right ]dK[2]-\int _1^0\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) K[1]}dK[1]\&\right ]\left [\int _1^{\frac {3}{8}}\frac {1}{(K[1]-1) K[1]}dK[1]-3 K[2]\right ]dK[2]\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE sqrt(-(-9*y(x) + 8*Derivative(y(x), (x, 2)))*y(x))/4 - 3*y(x)/4 + Derivative(y(x), x) cannot be solved by the factorable group method