Differential Equations Answers

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.

(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

Solve the following equation
sin(y')=0

x^2p^²-2xyp+2y^2-x^2=0
Using Claratiurs method

Solve (1+yz)dx+x(z-x)dy-(1+xy)dz=0

if z=ax+by+c the p=