Differential Equations Answers

Suppose the temperature of a body when discovered is 85^o F. Two hours later, the 

temperature is 74^o F and the room temperature is 68^o F. Find the time when the body was discovered after death (assume the body temperature to be 98.6^o F at the time of death.)

Find the value of n for which the equation (n - 1)² u_xx - y²ⁿ u_yy = ny^2n-1 u_y is parabolic or hyperbolic.

Identify the level curves of the following functions:

(i) √(x²+y²)

(ii) √(4 - x² + y²)

(iii) x-y

(iv) x/y

Find the differential equations of the space curve in which the two families of surfaces

u = x² - y² = c₁ and v = y² - z² = c₂ intersect.

Verify that the equations

i) z = √(2x + a) + √(2y+ b)

ii) z² + μ = 2 (1+λ^-1) (x+λy)

are both complete integrals of the PDE z = (1/p) + (1/q).

Also show that the complete integral (ii) is the envelope of the one parameter sub-system obtained by taking b = -( a/λ ) - ( μ/1+λ ) in the solution i).

Let x=e^r cosθ, y=e^r sinθ and f be a continuously differentible function of x and y, whose partial derivatives are also continuously differentible. show that ∂²f/∂r² + ∂²f/∂θ² = (x²+y²)(∂²f/∂x² + ∂²f/∂y²)

Using the method of undetermined coefficients, find the general solution of the DE

y^iv - 2y''' + 2y'' = 3e^-x + 2e^-x x + e^-x sinx

Identify the following differential equations and hence solve them

i) y' = -(4/x²) - (y/x) + y²

ii) y = xy' + 1 - ln y'

A certain population is known to be growing at a rate given by the logistic equation

dx/dt =x(b - ax)

Show that the minimum rate of growth will occur when the population is equal to half the

equilibrium size, that is, when the population is b / 2a .

do the functions y_{1 (t)=} √ t and y ₂ ( t ) =1\ t form a fundamental sets of solutions of the equation 2t ² y" + 3t y' - y=0, on the interval 0 <t< ∞ ? justify your answer.