Differential Equations Answers

(D^2-2DD'+D^2)z=12xy

Solve: e^(x)sinydx + (e^x +1)cosydy=0

(i) Use method of separation of variables in rectangular coordinates to obtain the following harmonic functions which are bounded in the upper half plane y > 0 of R2,
e−λy cosλx, e−λy sinλx; λ ≥0.
(ii) By integration with respect to the parameter λ, obtain the functions
x / x2 +y2, y / x2 +y2 which are harmonic in the upper half plane y > 0 of R2.

2. (i) Verify that
u(x,y;λ) = e−λy cosλx, −∞ < λ < ∞
is a one parameter family of solutions of Laplaces’s equation in R2.
(ii) Find v(x,y;λ) = ∂ ∂λu(x,y;λ) and verify that v(x,y;λ), − ∞ < λ < ∞, is also a one parameter family of solutions of Laplace’s equation in R2.
(iii) For (x,y) in the upper half plane y > 0, the improper integral
v(x,y) = ∞ u(x,y;λ)dλ 0
is convergent. Evaluate this integral and show by direct computation that v(x,y) is a solution of Laplace’s equation in the upper half plane.

Does principle of superposition also hold for non-linear PDEs? If not, then give an example of non-linear PDE where principle of superposition does not hold.

Let u satisfy the Laplace equation in a disk Ω = {(x,y) | x2 + y2 < 1} and continuous on ¯Ω. If u(cosθ,sinθ) ≤ sinθ + cos2θ, then show that
u(x,y) ≤ y +x2 −y2, ∀(x,y) ∈ Ω.

Apply the method of separation of variables to solve the following Dirichlet boundary value problem
uxx +uyy = 0, 0 <x<a, 0<y <b,
u(x,0) = f(x), u(x,b) = 0, 0 ≤ x ≤ a,
u(0,y) = 0, u(a,y) = 0, 0 ≤ y ≤ b.

Consider the Dirichlet problem in an annulus Ω in R3,
Ω={(r,θ) ∈ R2; 0 < a <r <1, −π ≤θ≤π}.
This problem requires one to find a function u(r,θ) in C2(Ω) ∩ C0(¯Ω) such that
∇^2 u(r,θ) = 0, a < r < 1, −π ≤θ ≤π
u(1,θ) = f(θ), −π ≤ θ ≤ π,
u(a,θ) = g(θ), −π ≤ θ ≤ π.
Assume that the solutions can be represented as a superposition of the functions 1,logr, rn cosnθ,rn sinnθ,r−n cosnθ,r−n sinnθ, n = 1,2,..., all of which are harmonic in Ω, and discuss the determination of the coefficients. Assume that f and g are C1 functions.

(y 2 +yz)dx+(xz+z 2 )dy+(y 2 −xy)dz=0 here P= (y^2+yz) , Q=(xz+z^2) , R=(y^2-xy)

Particular integral of d²y/dx² +3dy/xdx +2/x² = 0 is