Linear Higher Order Differential Equations

Refer to Differential Operator, Linear Dependence of Functions

General form #

$$ \begin{align*} a_{0}D^{n}y+a_{1}D^{n-1}y+a_{2}D^{n-2}y+…+a_{n-1}Dy+a_{n}y=g(x);\\ a_k(x)\\ In\\ short,\\ f(D)y=g(x) \end{align*} $$

If g(x)=0, then it’s called a homogeneous linear ODE.

Complementary Function of a Linear Higher Order ODE #

$$ C.F.\\ of\\ f(D)y=g(x)\\ is:\\ f(D)y=0 $$

existence? Theorem #

The initial value problem given by and n initial conditions [$y(x_0)=c_{0};y’(x_0)=c_{1}…D^{n}y(x_{0})=c_{n}$] where a_i(x) and g(x) are continuous in some interval I containing x₀ and if a₀≠0 in I, then the IVP has a unique solution defined throughout in I.

Solution #

Homogeneous #

The nth order linearly homogeneous ODE f(D)y=0 has n linearly independent solutions: $y_1,y_2,…y_n$, then the general solution of f(D)y=0 is:

$$ y_{c}(x)=c_{1}y_{1}+c_{2}y_{2}+…+c_{n}y_{n} $$

Non-homogeneous #

The nth order linearly non-homogeneous ODE f(D)y=g(x) has:

• $y_p(x)$: its particular solution ^a77c80
  • 0 arbitrary constants
• $y_c(x)$: general solution of the associated homogeneous equation $f(D)y=0$
  • n arbitrary constants

General solution of it is:

$$y_p(x)+y_c(x)$$

Associated general solution of C.F. + Particular solution of linear higher order ODE