Solution of general form of LDE
Solution of General Form of Linear Differential Equations (LDE)
Linear Differential Equations (LDEs) are a class of differential equations that are linear in the unknown function and its derivatives. The general form of an nth-order LDE can be written as:
$$ a_n(x)\frac{d^n y}{dx^n} + a_{n-1}(x)\frac{d^{n-1} y}{dx^{n-1}} + \cdots + a_1(x)\frac{dy}{dx} + a_0(x)y = g(x) $$
where $y$ is the unknown function of $x$, $a_0(x), a_1(x), \ldots, a_n(x)$ are coefficient functions of $x$, and $g(x)$ is the non-homogeneous term. If $g(x) = 0$, the equation is called homogeneous; otherwise, it is non-homogeneous.
Homogeneous vs Non-Homogeneous LDEs
| Aspect | Homogeneous LDE | Non-Homogeneous LDE |
|---|---|---|
| General Form | $L[y] = 0$ | $L[y] = g(x)$ |
| Non-Homogeneous Term | $g(x) = 0$ | $g(x) \neq 0$ |
| Principle of Superposition | Yes | No |
| Solution Structure | Composed of complementary function (CF) only | Composed of CF and particular solution (PS) |
Solution Method for Homogeneous LDEs
For a homogeneous LDE, the solution is called the complementary function (CF) and is obtained by solving the characteristic equation. The characteristic equation is derived by substituting $y = e^{rx}$ into the homogeneous LDE and simplifying.
Example: Second-Order Homogeneous LDE
Consider the second-order homogeneous LDE:
$$ a_2(x)\frac{d^2 y}{dx^2} + a_1(x)\frac{dy}{dx} + a_0(x)y = 0 $$
The characteristic equation is:
$$ a_2r^2 + a_1r + a_0 = 0 $$
Solving this quadratic equation for $r$ gives the roots, which determine the form of the CF:
- If roots are real and distinct, $r_1$ and $r_2$, then $CF = C_1e^{r_1x} + C_2e^{r_2x}$
- If roots are real and repeated, $r_1 = r_2 = r$, then $CF = (C_1 + C_2x)e^{rx}$
- If roots are complex, $r = \alpha \pm \beta i$, then $CF = e^{\alpha x}(C_1\cos(\beta x) + C_2\sin(\beta x))$
Solution Method for Non-Homogeneous LDEs
For a non-homogeneous LDE, the solution consists of two parts: the complementary function (CF) and the particular solution (PS). The CF is the solution to the corresponding homogeneous equation, while the PS is a specific solution that satisfies the non-homogeneous term.
Finding the Particular Solution
The method of undetermined coefficients or the method of variation of parameters can be used to find the PS.
Method of Undetermined Coefficients
This method is applicable when $g(x)$ is a polynomial, exponential, sine, cosine, or a combination of these. The form of the PS is guessed based on the form of $g(x)$, and coefficients are determined by substituting the PS into the LDE and equating coefficients.
Method of Variation of Parameters
This method is more general and can be used when the method of undetermined coefficients is not applicable. It involves finding functions $u_1(x), u_2(x), \ldots, u_n(x)$ such that:
$$ y_p = u_1y_1 + u_2y_2 + \cdots + u_ny_n $$
where $y_1, y_2, \ldots, y_n$ are solutions to the corresponding homogeneous equation. The functions $u_1(x), u_2(x), \ldots, u_n(x)$ are found by solving a system of equations derived from the original LDE.
Example: Second-Order Non-Homogeneous LDE
Consider the second-order non-homogeneous LDE:
$$ a_2(x)\frac{d^2 y}{dx^2} + a_1(x)\frac{dy}{dx} + a_0(x)y = g(x) $$
The solution is $y = CF + PS$, where CF is found as described above, and PS is found using one of the methods mentioned.
Summary
The solution of a general form of LDE involves the following steps:
- Determine if the LDE is homogeneous or non-homogeneous.
- For a homogeneous LDE, find the CF by solving the characteristic equation.
- For a non-homogeneous LDE, find the CF and then determine the PS using an appropriate method.
- The general solution is the sum of CF and PS for non-homogeneous LDEs.
Understanding the solution process for LDEs is crucial for solving a wide range of problems in mathematics, physics, engineering, and other fields where differential equations model dynamic systems.