One Dimensional Wave Equation is (A) 𝛿2y/𝛿t2 = α2 𝛿2y/𝛿x2 (B) 𝛿y/𝛿t = α2 𝛿2y/𝛿x2 (C) 𝛿2y/𝛿t2 = -𝛿2y/𝛿x2 (D) None of these

The Partial Differential equation is given as,

A 𝛿2u/𝛿x2 + B 𝛿2u/𝛿x𝛿y + C 𝛿2u/𝛿y2 + D 𝛿y/𝛿x + E 𝛿u/𝛿y = F

For the One-Dimensional equation,

α2 𝛿2y/𝛿x2 = 𝛿2y/𝛿t2

where, A = α , B = 0, C = -1

Put all the values in equation (1)

∴ 0 - 4 (α )(-1)

4α > 0.

So, this is a one-dimensional wave equation.

• 𝛿y/𝛿t = α2 𝛿2y/𝛿x2

Having A = α2, B = 0 and C = 0

Put all the values in equation (1), we get

0 - 4(α )(0) = 0, therefore it shows parabolic function.

So, this is a one-dimensional heat equation.

• 𝛿2y/𝛿t2 = -𝛿2y/𝛿x2

having A = 1, B = 0, C = 1

Put all the values in equation (1), we get

0 - 4(1)(1) = -4, therefore it shows elliptical function.

So, this is a two-dimensional heat equation.

Summary:

One Dimensional Wave Equation is (A) 𝛿2y/𝛿t2 = α2 𝛿2y/𝛿x2 (B) 𝛿y/𝛿t = α2 𝛿2y/𝛿x2 (C) 𝛿2y/𝛿t2 = -𝛿2y/𝛿x2 (D) None of these

𝛿2y/𝛿t2 = α2 𝛿2y/𝛿x2 is one-dimensional wave equation. A linear second-order partial differential equation that describes the propagation of oscillations at a fixed speed in some quantity.