Differential Equations Answers

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Determine the temperature distribution u(x, t) in a bar of length L, whose ends are kept at zero

temperature, and whose initial temperature is:

i) u(x, 0) = f(x) = U₀

ii) u(x, 0) = f(x) = k sin πx/L

(𝑥 − 𝑦)𝑦2𝑝 + (𝑦 − 𝑥)𝑥2𝑞 = (𝑥2 + 𝑦2)𝑧,

Find the Integra surface when 𝑥𝑧 = 𝑎3 (𝑎 > 0), 𝑦 = 0.

q-px-p^2=0

(xy²+ x -2y +3) dx ( x²y -2x -2y) dy =0

Solve dy/dx-4y=2x^2 using the method exploit the superstition.

The differential equations

dS/dt = - βSI + λS

dI/dt = βSI - γI

model a disease spread by contact, where S is the number of susceptibles, λ is the number

of infectives, β is the contact rate, γ is the removal rate and l is the birth rate of

susceptibles.

(i) Identify which term in the RHS of each differential equation arises from the birth of

susceptibles.

(ii) Discuss the model given by the above two differential equations.

A model corresponding to the cooperative interaction between two species x and y

is given by

dx/dt=(4-2x+y)x

dy/dt=(4+x-2y)y

Find all the equilibrium points of the system and discuss the stability of the system at

these points.

Using the method of undetermined coefficients, find the general solution of the DE y^iv -2y^m+2yⁿ=3e^-x+2e^-x+e^-xsin x.

write the three-dimensional Gaussian model dispersion for the atmospheric pollution

problem. Modify the model under the following assumptions:

(i) Wind velocity is in only y -direction.

(ii) Mass transfer due to bulk motion in the y -direction overshadows the

contributions due to mass diffusion.

(iii) Motion is in steady state.

(iv) Wind speed is constant.

(v) Diffusivities are constant in all the directions.