2.301 problem 878

Internal problem ID [9861]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 878
Date solved : Friday, October 11, 2024 at 12:00:56 PM
CAS classification : [_rational]

Solve

\begin{align*} y^{\prime }&=\frac {1+y^{4}-8 a x y^{2}+16 a^{2} x^{2}+y^{6}-12 y^{4} a x +48 y^{2} a^{2} x^{2}-64 a^{3} x^{3}}{y} \end{align*}

Unknown ode type.

2.301.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {1+y \left (x \right )^{4}-8 a x y \left (x \right )^{2}+16 a^{2} x^{2}+y \left (x \right )^{6}-12 y \left (x \right )^{4} a x +48 y \left (x \right )^{2} a^{2} x^{2}-64 a^{3} x^{3}}{y \left (x \right )} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {1+y \left (x \right )^{4}-8 a x y \left (x \right )^{2}+16 a^{2} x^{2}+y \left (x \right )^{6}-12 y \left (x \right )^{4} a x +48 y \left (x \right )^{2} a^{2} x^{2}-64 a^{3} x^{3}}{y \left (x \right )} \end {array} \]

2.301.2 Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying Chini 
differential order: 1; looking for linear symmetries 
trying exact 
Looking for potential symmetries 
trying inverse_Riccati 
trying an equivalence to an Abel ODE 
differential order: 1; trying a linearization to 2nd order 
--- trying a change of variables {x -> y(x), y(x) -> x} 
differential order: 1; trying a linearization to 2nd order 
trying 1st order ODE linearizable_by_differentiation 
--- Trying Lie symmetry methods, 1st order --- 
`, `-> Computing symmetries using: way = 2`[0, (-64*a^3*x^3+48*a^2*x^2*y^2-12*a*x*y^4+y^6+16*a^2*x^2-8*a*x*y^2+y^4-2*a+1)/
 

2.301.3 Maple dsolve solution

Solving time : 0.046 (sec)
Leaf size : 73

dsolve(diff(y(x),x) = (1+y(x)^4-8*a*x*y(x)^2+16*a^2*x^2+y(x)^6-12*y(x)^4*a*x+48*y(x)^2*a^2*x^2-64*a^3*x^3)/y(x), 
       y(x),singsol=all)
 

\[ -\left (\int _{\textit {\_b}}^{y}\frac {\textit {\_a}}{\textit {\_a}^{6}-12 \textit {\_a}^{4} a x +48 \textit {\_a}^{2} a^{2} x^{2}-64 a^{3} x^{3}+\textit {\_a}^{4}-8 \textit {\_a}^{2} a x +16 a^{2} x^{2}-2 a +1}d \textit {\_a} \right )+x -c_{1} = 0 \]

2.301.4 Mathematica DSolve solution

Solving time : 0.245 (sec)
Leaf size : 130

DSolve[{D[y[x],x] == (1 + 16*a^2*x^2 - 64*a^3*x^3 - 8*a*x*y[x]^2 + 48*a^2*x^2*y[x]^2 + y[x]^4 - 12*a*x*y[x]^4 + y[x]^6)/y[x],{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [2 a \left (x-\frac {1}{2} \text {RootSum}\left [64 \text {$\#$1}^3 a^3-48 \text {$\#$1}^2 a^2 y(x)^2-16 \text {$\#$1}^2 a^2+12 \text {$\#$1} a y(x)^4+8 \text {$\#$1} a y(x)^2+2 a-y(x)^6-y(x)^4-1\&,\frac {\log (x-\text {$\#$1})}{48 \text {$\#$1}^2 a^2-24 \text {$\#$1} a y(x)^2-8 \text {$\#$1} a+3 y(x)^4+2 y(x)^2}\&\right ]\right )=c_1,y(x)\right ] \]