Test: Method of Separation of Variables - Civil Engineering (CE) MCQ

⇒ from the Integrating the equation, we get

⇒ at x = 0, y = 1, hence -1/1 = c ⇒ c = -1

x ≠ 1, at (x = 1) y will be undefined

x > 1, x < 1

The given equation is in the form of

M dx + N dy = 0

Here M = y e^xy

N = (x e^xy + 2y)

 

Hence the differential equation is an exact equation.

The solution is

Concept:

Separation of Variables:

Calculation:

Given:

Differential equation:

By separation of variables:

Integrating on both sides we will get:

3yy’ + 4x = 0

3ydy = - 4x dx

On integration both the sides:

Where c is the constant of integration.

Thus, the solution of given differential equation represents the family of Ellipses.

Given differential equation is,

dx – (x + y + 1) dy = 0

Put (x + y + 1) = t

Now, the differential equation becomes


⇒ t - In (t + 1) = x + C₁

⇒ (x + y + 1) – In (x + y + 2) = x + C₁

⇒ y + 1 – C₁ = In (x + y + 2)

⇒ In (x + y + 2) = y + k

⇒ (x + y + 2) = e^y+k

⇒ (x + y + 2) e^-y = C

The given differential equation is, (dy/dx) = ky

dy/y = kdx

On integrating both the sides, we get

ln y = kx + ln c

y = ce^kx

Given differential equation is,

By integrating both sides,

dy/dx = x/y

⇒ ydy = xdx

By integrating on both sides, we get

Given the initial condition, x = 0, y = 1.

⇒ 1/2 = 0 + C

⇒ C = 1/2

Now, the solution becomes

y² = x² + 1

Given the differential equation is,


By integrating with respect to ‘x’, we get

⇒ ln(20) = C

Now, the differential equation becomes

At x = 2, y = 20 (e² + 1) = 167.78

The method of separation of variables is used to solve a wide range of linear partial differential equations with boundary and initial conditions, such as:

• Heat equation
• Wave equation
• Laplace equation
• Helmholtz equation
• Biharmonic equation