Question

For simultaneous equations in x and y, if D=60, Dx=120, Dy=-540, then what is the value of x & y?​

Answer 1

Answer:

Solutions of Linear

Differential Equations

A . l Linear Differential Equations with

Constant Coefficients

Linear diflFerential equations with constant coefficients are usually writ-

ten as

2/("> + ai2/("-i) + ... + a„_i2/(i) + anV = g, (A.l)

where a^, fc = 1,..., n, are numbers, y^^^ = ^ , and g = g{t) is a known

function of t. We shall denote hy D = ^ the derivative operator^ so that

the differential equation now becomes

p{D)y = (D^ + aiD^-i + ... + a^_iD + an)y = g. (A.2)

If g(t) = 0, the equation is said to be homogeneous. If g{t) ^ 0, then the

homogeneous or reduced equation is obtained from (A.2) by replacing g

byO.

If y and y* are two different solutions of (A.2), then it is easy to

show that y — y* solves the reduced equation of (A.2). Hence, if y is any

solution to (A.2), it can be written as

y = y*+y\ (A.3)

where y* is any other particular solution to (A.2) and y^ is a suitable

solution to the homogeneous equation. Therefore, solving (A.2) involves

(a) finding all the solutions to the homogeneous equation, caUed the gen-

eral solution, and (b) finding a particular solution to the given equation.