1.507 problem 510
Internal
problem
ID
[9489]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
510
Date
solved
:
Thursday, October 17, 2024 at 03:44:10 PM
CAS
classification
:
[`y=_G(x,y')`]
Solve
\begin{align*} x^{2} \left (x^{2} y^{4}-1\right ) {y^{\prime }}^{2}+2 x^{3} y^{3} \left (y^{2}-x^{2}\right ) y^{\prime }-y^{2} \left (x^{4} y^{2}-1\right )&=0 \end{align*}
Solving for the derivative gives these ODE’s to solve
\begin{align*}
\tag{1} y^{\prime }&=-\frac {\left (x^{2} y^{4}-x^{4} y^{2}-\sqrt {y^{8} x^{4}-y^{6} x^{6}+y^{4} x^{8}-x^{2} y^{4}-x^{4} y^{2}+1}\right ) y}{\left (x^{2} y^{4}-1\right ) x} \\
\tag{2} y^{\prime }&=-\frac {\left (x^{2} y^{4}-x^{4} y^{2}+\sqrt {y^{8} x^{4}-y^{6} x^{6}+y^{4} x^{8}-x^{2} y^{4}-x^{4} y^{2}+1}\right ) y}{\left (x^{2} y^{4}-1\right ) x} \\
\end{align*}
Now each of the above is solved
separately.
Solving Eq. (1)
Solving Eq. (2)
1.507.1 Maple step by step solution
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (y \left (x \right )^{4} x^{2}-1\right ) \left (\frac {d}{d x}y \left (x \right )\right )^{2}+2 x^{3} y \left (x \right )^{3} \left (y \left (x \right )^{2}-x^{2}\right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )^{2} \left (y \left (x \right )^{2} x^{4}-1\right )=0 \\ \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} \\ {} & {} & \left [\frac {d}{d x}y \left (x \right )=\frac {\left (-y \left (x \right )^{4} x^{2}+y \left (x \right )^{2} x^{4}+\sqrt {y \left (x \right )^{8} x^{4}-y \left (x \right )^{6} x^{6}+y \left (x \right )^{4} x^{8}-y \left (x \right )^{4} x^{2}-y \left (x \right )^{2} x^{4}+1}\right ) y \left (x \right )}{\left (y \left (x \right )^{4} x^{2}-1\right ) x}, \frac {d}{d x}y \left (x \right )=-\frac {\left (y \left (x \right )^{4} x^{2}-y \left (x \right )^{2} x^{4}+\sqrt {y \left (x \right )^{8} x^{4}-y \left (x \right )^{6} x^{6}+y \left (x \right )^{4} x^{8}-y \left (x \right )^{4} x^{2}-y \left (x \right )^{2} x^{4}+1}\right ) y \left (x \right )}{\left (y \left (x \right )^{4} x^{2}-1\right ) x}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=\frac {\left (-y \left (x \right )^{4} x^{2}+y \left (x \right )^{2} x^{4}+\sqrt {y \left (x \right )^{8} x^{4}-y \left (x \right )^{6} x^{6}+y \left (x \right )^{4} x^{8}-y \left (x \right )^{4} x^{2}-y \left (x \right )^{2} x^{4}+1}\right ) y \left (x \right )}{\left (y \left (x \right )^{4} x^{2}-1\right ) x} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=-\frac {\left (y \left (x \right )^{4} x^{2}-y \left (x \right )^{2} x^{4}+\sqrt {y \left (x \right )^{8} x^{4}-y \left (x \right )^{6} x^{6}+y \left (x \right )^{4} x^{8}-y \left (x \right )^{4} x^{2}-y \left (x \right )^{2} x^{4}+1}\right ) y \left (x \right )}{\left (y \left (x \right )^{4} x^{2}-1\right ) x} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]
1.507.2 Maple trace
`Methods for first order ODEs:
-> Solving 1st order ODE of high degree, 1st attempt
trying 1st order WeierstrassP solution for high degree ODE
trying 1st order WeierstrassPPrime solution for high degree ODE
trying 1st order JacobiSN solution for high degree ODE
trying 1st order ODE linearizable_by_differentiation
trying differential order: 1; missing variables
trying simple symmetries for implicit equations
Successful isolation of dy/dx: 2 solutions were found. Trying to solve each resulting ODE.
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying homogeneous types:
trying exact
Looking for potential symmetries
trying an equivalence to an Abel ODE
trying 1st order ODE linearizable_by_differentiation
-> Solving 1st order ODE of high degree, Lie methods, 1st trial
`, `-> Computing symmetries using: way = 3
`, `-> Computing symmetries using: way = 4
`, `-> Computing symmetries using: way = 2
`, `-> Computing symmetries using: way = 2
-> Solving 1st order ODE of high degree, 2nd attempt. Trying parametric methods
trying dAlembert
-> Calling odsolve with the ODE`, diff(y(x), x) = (-y(x)^4*RootOf(2*x*y(x)^3*_Z^5+(-y(x)^4*x^2+y(x)^4)*_Z^4-2*x*y(x)^5*_Z^3+_Z^2*x^2
Methods for first order ODEs:
--- Trying classification methods ---
trying homogeneous types:
trying exact
Looking for potential symmetries
trying an equivalence to an Abel ODE
trying 1st order ODE linearizable_by_differentiation
-> Calling odsolve with the ODE`, diff(y(x), x) = (y(x)^4*RootOf(2*x*y(x)^3*_Z^5+(y(x)^4*x^2-y(x)^4)*_Z^4-2*x*y(x)^5*_Z^3+_Z^2-x^2*y
Methods for first order ODEs:
--- Trying classification methods ---
trying homogeneous types:
trying exact
Looking for potential symmetries
trying an equivalence to an Abel ODE
trying 1st order ODE linearizable_by_differentiation
-> Solving 1st order ODE of high degree, Lie methods, 2nd trial
`, `-> Computing symmetries using: way = 4
`, `-> Computing symmetries using: way = 5
`, `-> Computing symmetries using: way = 5`
1.507.3 Maple dsolve solution
Solving time : 3.244
(sec)
Leaf size : maple_leaf_size
dsolve(x^2*(y(x)^4*x^2-1)*diff(y(x),x)^2+2*x^3*y(x)^3*(y(x)^2-x^2)*diff(y(x),x)-y(x)^2*(x^4*y(x)^2-1) = 0,
y(x),singsol=all)
\[ \text {No solution found} \]
1.507.4 Mathematica DSolve solution
Solving time : 0.0
(sec)
Leaf size : 0
DSolve[{-(y[x]^2*(-1 + x^4*y[x]^2)) + 2*x^3*y[x]^3*(-x^2 + y[x]^2)*D[y[x],x] + x^2*(-1 + x^2*y[x]^4)*D[y[x],x]^2==0,{}},
y[x],x,IncludeSingularSolutions->True]
Not solved