Differential Equations Answers

A circuit consists of a resistance R ohms and an inductance of L henry connected to a

generator of E cos(ωt + α) voltage. Find the current in the circuit. (i = 0, when t = 0).

Find power series method to find the solution of the given differential equation about the

ordinary point x = 0.

y′′ + e^xy′ − y = 0

1a. Show from first principles, i.e., by using the definition of linear independence,

that if μ = x + iy, y ̸= 0 is an eigenvalue of a real matrix

A with associated eigenvector v = u + iw, then the two real solutions

Y(t) = eat(u cos bt − wsin bt)

and

Z(t) = eat(u sin bt + wcos bt)

are linearly independent solutions of ˙X = AX

1b.Use (a) to solve the system

˙X =

(

3 1

−8 7

)

X.

NB: Real solutions are required.

Reduce the system

(D2 + 1)[x] − 2D[y] = 2t

(2D − 1)[x] + (D − 2)[y] = 7.

to an equivalent triangular system of the form

P1(D)[y] = f1(t)

P2(D)[x] + P3(D)[y] = f2(t)

and solve.