- #1
cefarix
- 78
- 0
How would I solve the differential equation y' = - y / ( (x^2 + y^2)^1.5 ) ? Btw, its not a homework prob... thanks
can we replace y' by tan theta?
cefarix said:Btw, its not a homework prob... thanks
cefarix said:[tex]\frac{dy}{dx} = - \frac{r \cos \theta + \frac{dr}{d\theta} \sin \theta}{r \sin \theta + \frac{dr}{d\theta} \cos \theta}[/tex]
Tom Mattson said:The idea behind moving homework questions to the Homework section is to clear up the Math section for discussion of mathematical topics, without having this kind of step-by-step help discussions hanging around. While this may not be a homework problem that was assigned to you, it looks just like one, and it is handled just like one, so it really should be posted there.
Hurkyl said:It doesn't look like the x-axis is the asymptote for those plots.
I don't really know if this approach is kosher, but...
I made the change of variable:
y → y(t) / z(t)
x → x(t) / w(t)
from which I get the differential equation:
[tex]
(y' z - y z') (x^2 z^2+ y^2 w^2)^{3/2} = (x'w - xw') z^4 y w
[/tex]
where differentiation is with respect to t.
Now, notice that when w(a) = 0 (corresponding to infinite x in the original problem), I can satisfy the differential equation
y'z - yz' = 0
Or equivalently,
y = Kz
meaning that the original y(t) is roughly the constant K when t is near a.
cefarix said:Can someone show me a particular solution of the two equations y' = -y/((x^2+y^2)^1.5) and x' = -x/((x^2+y^2)^1.5)?
If you're not satisfied with that, do the polar coordinate substitution of the dependent variables.cefarix said:I'm not sure if dy/dx would be the correct form, because then I wouldn't be able to find out the time-dependent equation.
A differential equation is a mathematical equation that expresses the relationship between a function and its derivatives. It involves variables, constants, and functions that are related to each other by derivatives.
The purpose of a differential equation is to model and solve real-world problems that involve rates of change. Differential equations are used in various fields such as physics, engineering, economics, and biology to describe the behavior of systems over time.
The method for solving a differential equation depends on its type. There are several techniques, such as separation of variables, integrating factors, and substitution, that can be used to solve different types of differential equations. In some cases, a differential equation may also be solved numerically using computer software.
Differential equations have numerous applications in various fields of science and engineering. They are used to model physical phenomena such as motion, heat transfer, and population growth. In engineering, differential equations are used to design and analyze systems such as circuits, control systems, and structures.
Yes, there are several types of differential equations, including ordinary differential equations, partial differential equations, and stochastic differential equations. These types differ in the number of independent variables, the types of functions involved, and the methods used to solve them.