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).
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)
Z(t) = eat(u sin bt + wcos bt)
are linearly independent solutions of ˙X = AX
1b.Use (a) to solve the system
NB: Real solutions are required.