Differential Equations Answers

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

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

4. (a)Find the general solution of the partial differential equation.
zxy =(6x^3)y

(b) Find the particular solution for which z(x,0) = 6x^2 , z(1, y) = 6cos y

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.

Solution of (D^3 -D)y = x^3 is

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.

State whether the following statements are true or false. Justify yourself with the help of a short proof or a counter example.
i) y′ + P(x) y = Q(x) y^n is a linear equation for integral values of n.

ii) y = 0, is a singular solution of the differential equation 27y-8(dy/dx)^3=0

iii) Equation x^2 ( y − px) = yp^2 is reducible to clairaut’s form

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)not=to (0,0)
0 (x,y)= to (0,0). }

Is fx continuous at (0,0) ? Justify your answer.

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.

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 .