Differential Equations Answers

Cersei is the ruler of a country and her army is fighting a war. The probability of her side winning the war depends upon two variables - the average fitness of her army (given by x) and the morale of her army (denoted by m). Furthermore, both the fitness and morale are functions of how much food is available to her army. If f kilogram of food is available to her army (per person per day), then x(f) = −f 2 + 5f and m(f) = f − 1 (where f ∈ [0, 5]). The probability of her side winning the war is given by a function of x and m and is given by the function g such that g(x, m) = x 2+2m+2 40 . Currently, her army is getting 1.5 kg of food per person per day. By approximately how much will her army’s probability of winning change if she increases the supply of food by a very small amount?

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

Compute fxy and fyx for the function f(x,y) = e^(x+y) +9x^2 +2xy at (1,2).

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.

Check whether the function f(x,y)= { 4x^2y/4x^4+y^2,. (x,y)not=to(0,0) 0 (x,y) = to (0,0)} is continuous at ( 0,0).

Find the two repeated limits of the function f(x ,y) = (y-x/y+x) (1+x^2)/1+y^2) at (0,0) . Does the simultaneous limit of f exist as (x, y) →(0,0) ? Give reasons for your answer.

Show that the closed sphere with centre (2,3,7)and radius 10 in R^3 is contained in the open cube P = {(x, y,z): |x − 2 |<11, |y − 3| <11, |z − 7| <11}.

Show that the closed sphere with centre (2,3,7) 3and radius 10 in 3 R is contained in the open cube P = {(x, y,z :) x − 2 <11, y − 3 <11, z − 7 <11}.

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.