Differential Equations Answers

Find the cylindrical coordinates of the points where the Cartesian coordinates are

i)( 6,6,8)
ii) ( √2,1,1)

Solve the differential equation :
(x^2)y'' - y' - 4(x^3)y = 8(x^3)sin(x^2)

Consider the following system of differential equations representing a prey and predator
population model
x y
dt
dx
= −
2

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

Formulate a one-dimensional model describing the dynamics of phytoplankton growth C(x, t)
in a water mass taking into account the following: D , its diffusion coefficient, α its rate of
growth, β its mortality rate due to sinking. Fixing the area of interest as 0 ≤ x ≤1 and the initial
concentration of phytoplankton as 3
20 moles cm/ , find the concentration distribution of
phytoplankton in 0 ≤ x ≤1 at any time t .

write the ordinary differential equation (1+sin y)dx = (2ycosy-x(secy+tan y))dy

Using the method of undermined coefficients, find the general solution of the differential equation y^iv-2y'''+2y''=3e^-x+2e^-xx+e^-x sin x

Find the solution of the Riccati equation
dy/dx=(2cos^2(x)-sin^2(x)+y^2)/2cos(x); y1(x)=sinx

Find the value of b for which the equation
( ye^(2xy)+x)dx+bxe^(2xy)dy= 0
is exact, and hence solve it for that value of b

Solve: z(p-q) = z^2 + (x+(y^2))

A series RLC circuit with R = 6 ohm, C = 0.02 Farad and L = 0.1 has no applied voltage. Find the subsequent current in the circuit if the initial charge, on the capacitor is q0 and the initial current is zero.