Differential Equations Answers

Solve the differential equation x^(3)p^(2)+x^(2)yp+a^(3)=0 and also obtain its singular solution if

Using general solution of a tightly stretched square membrane of side 2 units having wave velocity
c = 2, find the deflection, if initial velocity is 0 and initial difference is 4sin2πxsin3πy.

State whether the following statements are true or false. Justify your answer with the help of a short proof or a counter example. i) Equation cos(x+y)p + sin(x+y)q = z^2 is a quasi-linear equation. ii) The solution of PDE dz/dx + dz/dy =z^2 is z = -[y+f(x-y)] Iii)(dz/dx)(dz/dy) - (dz/dy)^2 = 0 is a non-linear PDE
Please do it stepwise as I am a noob in differential equations

Consider the following system of differential equations representing a prey and predator
population model

dx/dt=x square -y

dy/dt= x+y
i) Identify all the real critical points of the system
ii) Obtain the type and stability of these critical points.

given that y1(x) = x^-1 is one solution of the differential equation 2x^2 dy^2/dx^2 + 3xdy/dx - y=0, x>0 find a second linearly independent solution of the equation

find fx(0,0) and fx (x,y), where (x,y)≠(0,0) for the following function f(x,y)= {xy^3/(x^2+y^2), (x,y) ≠(0,0) 0, (x,y)=(0,0) is fx continuous at (0,0)?

1. Using u=XY solve the boundary valued problem

du/dx + 2du/dy = 4u

u(x,0) = 2e^3x + 4e^5x

3. Find the general solution of the partial differential equation

6uxx −5uxy −6uyy =52e^3x+2y

A bacterial population is known to have a logistic growth pattern with initial population 1000 and an equilibrium population of 10,000. A count shows that at the end of 1 hr there are 2000 bacteria present. Determine the population as a function of time.

y"−3y' + 2y = 3e−x−10cos3x, y(0) = 1, y'(0) = 2.