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Classify the equation y 2uxx − uxy − y 2ux − uyy − x 2uy = 0, and find the characteristics curves that will reduce the equation to canonical form. (You do not need to reduce to canonical form

Find complete integral of 2p1x2x3 + 3p2x3^2 + P2 ^2 P3 = 0

An RCL circuit connected in series has R = 10 ohms, C = 10^2 farad, L = 1.5 henry, and an applied voltage E = 12 volts. Assuming no initial current and no initial charge at t = 0 when the voltage is first applied, find the subsequent current in the system.

Solve the equation (𝑥 + 𝑠𝑖𝑛𝑦)𝑑𝑥 + (𝑦2 + 𝑥𝑐𝑜𝑠𝑦)𝑑𝑦 = 0

Find the orthogonal trajectories of the following curves (where a is a parameter). r = a(1 + sin θ)

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).

(x^3+y^3)=(3xy^2)dy/dx

Use the Laplace transform to solve the given initial-value problem.

y′ + 2y = sin 4t, y(0) = 1

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.

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