83.34.6 problem 6

Internal problem ID [19348]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (H) at page 118
Problem number : 6
Date solved : Monday, March 31, 2025 at 07:09:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.062 (sec). Leaf size: 32

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

\[ y = \operatorname {RootOf}\left (-\ln \left (x \right )+c_2 -\int _{}^{\textit {\_Z}}\frac {1}{{\mathrm e}^{\textit {\_f}} c_1 +4 \textit {\_f} +2}d \textit {\_f} \right ) x^{2} \]

✓ Mathematica. Time used: 0.228 (sec). Leaf size: 386

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

\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{-e^{\frac {K[2]}{x^2}} c_1 x^2-e^{\frac {K[2]}{x^2}} \int _1^{\frac {K[2]}{x^2}}2 e^{-K[1]} K[1]dK[1] x^2+2 K[2]}dK[2]-\int _1^x\left (\frac {K[3] \left (e^{\frac {y(x)}{K[3]^2}} c_1+e^{\frac {y(x)}{K[3]^2}} \int _1^{\frac {y(x)}{K[3]^2}}2 e^{-K[1]} K[1]dK[1]\right )}{-e^{\frac {y(x)}{K[3]^2}} c_1 K[3]^2-e^{\frac {y(x)}{K[3]^2}} \int _1^{\frac {y(x)}{K[3]^2}}2 e^{-K[1]} K[1]dK[1] K[3]^2+2 y(x)}+\int _1^{y(x)}-\frac {\frac {4 K[2]^2}{K[3]^3}+\frac {2 e^{\frac {K[2]}{K[3]^2}} \int _1^{\frac {K[2]}{K[3]^2}}2 e^{-K[1]} K[1]dK[1] K[2]}{K[3]}+\frac {2 e^{\frac {K[2]}{K[3]^2}} c_1 K[2]}{K[3]}-2 e^{\frac {K[2]}{K[3]^2}} c_1 K[3]-2 e^{\frac {K[2]}{K[3]^2}} K[3] \int _1^{\frac {K[2]}{K[3]^2}}2 e^{-K[1]} K[1]dK[1]}{\left (-e^{\frac {K[2]}{K[3]^2}} c_1 K[3]^2-e^{\frac {K[2]}{K[3]^2}} \int _1^{\frac {K[2]}{K[3]^2}}2 e^{-K[1]} K[1]dK[1] K[3]^2+2 K[2]\right ){}^2}dK[2]\right )dK[3]=c_2,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**2 + sqrt(4*x**4*Derivative(y(x), (x, 2)) + x**4 + 16*y(x)**2))/(2*x) cannot be solved by the factorable group method