- #1
nrhoades
- 11
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Is there an analytic solution for:
y"(c+dx+ex^2) + ay + b = 0,
y (x=0) = Ts
y'(x=L) = 0
where a,b,c,d,e are all constants?
y"(c+dx+ex^2) + ay + b = 0,
y (x=0) = Ts
y'(x=L) = 0
where a,b,c,d,e are all constants?
I don't understand this DE. Are you saying that the second derivative of y, evaluated at c+dx+ex^2, is equal to -ay(x) - b, where y is now evaluated at x? What physical mechanism could give such a remote relationship? What about the domain? Over what range of x is this equation supposed to hold? Are c, d, and e such that c + dx + ex^2 precisely maps this range to itself? If not, how do we deal with the regions where y''(c+dx+ex^2) is defined but y(x) isn't, or vice versa?nrhoades said:Is there an analytic solution for:
y"(c+dx+ex^2) + ay + b = 0,
y (x=0) = Ts
y'(x=L) = 0
where a,b,c,d,e are all constants?
A non-homogeneous differential equation is one where the right-hand side of the equation contains a function that is not equal to zero. This function is typically referred to as the non-homogeneous or forcing function.
The solution to a non-homogeneous differential equation involves finding both the general solution to the equation and a particular solution that satisfies the non-homogeneous portion of the equation. The general solution is found by solving the associated homogeneous equation, while the particular solution is found using a method such as variation of parameters or undetermined coefficients.
Yes, a non-homogeneous differential equation can have multiple solutions. This is because the general solution to the equation can have an arbitrary constant or constants, resulting in a family of solutions. The particular solution, however, will be unique.
Non-homogeneous differential equations are commonly used in physics, engineering, and economics to model systems with external forces or inputs. Some examples include the motion of a damped harmonic oscillator with a driving force, the growth of a population with a limiting factor, and the change in value of an investment with added interest or inflation.
Yes, there are several numerical methods for solving non-homogeneous differential equations, including Euler's method, the Runge-Kutta method, and the finite difference method. These methods involve approximating the solution at discrete points and using iterative calculations to find an approximate solution to the equation.