Differential Equations Answers

A rectangular steel sheet is bounded by the axis x = 0, y = 0, x = a and y = b. The temperature along the edge x = 0 are kept at 100°C and other edges are at 0°C. Let u(x,y) denote the temperature satisfying the equation [15] + d^2u/dx^2 + d^2u/dy^2 = 0 Find the steady state temperature u(x, y), by assuming the solution to be of the form u(x, y) = (AePX + Be-px)(C cospy + D sin py).

Let L1 be the line in R3 with equation (x,y,z)=(1,0,2)+t(−1,3,4); t∈R

and let L2 be the line that is parallel to L1 and contains the point (1, −1, 3). Let V be the plane that contains both the lines L1 and L2.

(a) Find two vectors that are both parallel to the plane V but are not parallel to one another.

(b) Find a vector that is perpendicular to the plane V .

(c) Find an equation for the plane V .

(d) Find an equation for the line L3 that is perpendicular to the plane V and contains the point (1, −1, 4) .

Hint: Find a parametric equation for L3. Don’t try to find a Cartesian equation for L3.

Given the function

𝑔(𝑥) = 𝑥 𝐽1 (𝑥) − 𝐴 𝐽0(𝑥) with 𝐴 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ≥ 0,

determine 𝑔′, 𝑔 ′′ and 𝑔′′′ . Reduce all expressions to functions of 𝐽0(𝑥), 𝐽1(𝑥) and 𝐴 only.

(D²+7DD'+12D'²)z=sinhx

y2 dx — x(2x + 3y)dy = 0

Find the general solution of y"-y'-2y=4x²

Find dy/dx and simply the result, if possible.(with solution)

A.     y=√x -(1/√x)

B.     y=x^2+π^2+x^π

C.     y=(sin x-1)/(cos x)

D.     y=x^2 sec x

E.      y=(1)/(e^× +2)

Solve IBVP

∂u/∂t (x,t)=2 (∂^2 u)/(∂x^2 )+x+2,0<x<1,t>0

u(x,0)=x+2,0<x<1,u(0,t)=2,u(2,t)=-2

(𝒚 𝟐 + 𝒙𝒚𝟑 ) 𝒅𝒙 + (𝟓𝒚 𝟐 − 𝒙𝒚 + 𝒚 𝟐 𝐬𝐢𝐧 𝒚) 𝒅𝒚 = 𝟎