23.2.185 problem 190

Internal problem ID [5540]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 190
Date solved : Tuesday, September 30, 2025 at 12:51:45 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} 3 x^{4} {y^{\prime }}^{2}-x y-y&=0 \end{align*}

✓ Maple. Time used: 0.459 (sec). Leaf size: 201

ode:=3*x^4*diff(y(x),x)^2-x*y(x)-y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \frac {\left (\operatorname {arctanh}\left (\sqrt {x +1}\right ) \sqrt {3}\, x \sqrt {x +1}+3 c_1 x \sqrt {x +1}+\sqrt {3}\, x +\sqrt {3}\right )^{2}}{36 \left (x +1\right ) x^{2}} \\ y &= \frac {\left (\operatorname {arctanh}\left (\sqrt {x +1}\right ) \sqrt {3}\, x \sqrt {x +1}-3 c_1 x \sqrt {x +1}+\sqrt {3}\, x +\sqrt {3}\right )^{2}}{36 \left (x +1\right ) x^{2}} \\ y &= \frac {\left (\operatorname {arctanh}\left (\sqrt {x +1}\right ) \sqrt {3}\, x \sqrt {x +1}-3 c_1 x \sqrt {x +1}+\sqrt {3}\, x +\sqrt {3}\right )^{2}}{36 \left (x +1\right ) x^{2}} \\ y &= \frac {\left (\operatorname {arctanh}\left (\sqrt {x +1}\right ) \sqrt {3}\, x \sqrt {x +1}+3 c_1 x \sqrt {x +1}+\sqrt {3}\, x +\sqrt {3}\right )^{2}}{36 \left (x +1\right ) x^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.096 (sec). Leaf size: 171

ode=3 x^4 (D[y[x],x])^2-x y[x]-y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x^2 \text {arctanh}\left (\sqrt {x+1}\right )^2+2 x \text {arctanh}\left (\sqrt {x+1}\right ) \left (\sqrt {x+1}-\sqrt {3} c_1 x\right )+3 c_1{}^2 x^2+x-2 \sqrt {3} c_1 x \sqrt {x+1}+1}{12 x^2}\\ y(x)&\to \frac {x^2 \text {arctanh}\left (\sqrt {x+1}\right )^2+2 x \text {arctanh}\left (\sqrt {x+1}\right ) \left (\sqrt {x+1}+\sqrt {3} c_1 x\right )+3 c_1{}^2 x^2+x+2 \sqrt {3} c_1 x \sqrt {x+1}+1}{12 x^2}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 21.545 (sec). Leaf size: 728

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

\[ \text {Solution too large to show} \]