Linear Differential Equations
Linear Differential Equation of nth Order
Linear differential equation is in the form of Ly=f, where ‘L’ is a linear operator, ‘Y’ is a unknown function and ‘F’ is a known function of a same nature. In mathematical point of view first-order linear differential equation are those equation that can be kept in form.
1. A differential equation of the form
where are constant and Q in any function of x or constant is called linear differential equating of nth order with constant co-effieient.
(2) are equation of second order.
3.Solution f the equation f (D) y = Q
If are independent solutions of the equation , or f (D)y = 0 then, will also be solution of the equation.
If be a general solution of the equation f (D)Y = 0 and be particular solution of f (D)Y = Q then is the general solution of f (D)y = Q.
4.If m replace D by m then the equation
reduces to but
So, (i) is called auxiliary equation. Equation (i) is quadratic in m and must have two roots.
Case 1: auxiliary equation having different and real roots, equal roots, imaginary roots.
5. Some useful relations to find P.I.
where m is positive integer
In this case we have to use binomial theorem and then find derivatives successively, for this we need following relations.
, v is function of x or constant.
v is function of x.
[=real part of is [co-efficient of I in
10. Homogenous Linear equation
An equation of the form where are constant and Q is function of x, is called homogenous linear equation
is homogeneous linear equation of second order.
11. Equation Reducible to the Homogeneous Linear Form.
The equation of the form
is reduced to the homogeneous linear form by putting;
A + bx = t so that the equation can be directly reduced to one with constant co-efficient by putting
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