- #1
asdf1
- 734
- 0
another O.D.E. question:
xyy`= 2y^2 +4x^2
also unseparable...
xyy`= 2y^2 +4x^2
also unseparable...
Galileo said:whenever you have y'=F(y/x), the equation also called homogeneous (different from the homogeneous linear ODE definition), this one is invariant under 'zoom'.
Whoopsie, you've lost an x.asdf1 said:hmmm... here's my work~
(i forgot to include that the original condition is y(2)=4, sorry about that!)
y`= 2y/x+4x/y
suppose y=vx
=>v=y/x
so y`=v+v`x
=> v+xv`=2v +4/v
=> v`x=v+4/v
=> v`x=(v^2 +4)/v
=> v/(v^2 +4)dv=dx
The actual answer also needs a square root since you'll get a relation between y^2 and x. Now if this equation is true both the positive and negative square roots must be valid solutions to the ODE, but a little bit of thought shows that no more than one can fit the same boundary condition (y(2)=4)now my problem is that when you take the square root on both sides of the equations, the right side should have a positive and negative root, but the correct answer should only have a positive square root~
why?
A linear DE of, say, second order:btw, what is meant by the different homogeneous definitions and "zoom"?
An unseparable O.D.E (ordinary differential equation) is a type of differential equation where the dependent variable and its derivative cannot be separated on different sides of the equation. This means that the equation cannot be solved using traditional separation of variables methods.
The main difference between separable and unseparable O.D.E.s is in the way their variables can be separated. In a separable O.D.E, the dependent variable and its derivative can be isolated on different sides of the equation, making it easier to solve. In an unseparable O.D.E, the dependent variable and its derivative cannot be separated, making it more difficult to solve.
To solve an unseparable O.D.E, you can use a variety of methods such as substitution, integration by parts, or power series. It is important to carefully examine the equation and try different methods to find the most suitable approach.
Some common techniques used to solve unseparable O.D.E.s include substitution, integration by parts, and power series. These methods involve manipulating the equation to make it separable or using advanced mathematical techniques to find a solution.
Unseparable O.D.E.s are important in science because they arise in many real-world situations, particularly in physics and engineering. By solving these equations, scientists and engineers can better understand and predict the behavior of complex systems. Additionally, the techniques used to solve unseparable O.D.E.s can be applied to other types of mathematical problems.