Question

The solution of `dy/dx` − y = 1, y(0) = 1 is given by ______.

विकल्प

• y = −e^x + 1

• y = −e ^{− x − 1}

• y = −1 + ex

• y = 2e^х − 1

Answer 1

The solution of `dy/dx` − y = 1, y(0) = 1 is given by y = 2e^х − 1.

Explanation:

The equation `dx/dy` − y = 1 is a linear differential equation. rearranging it as `dy/(1 + y) = dx` and integrating both sides gives:

In(1 + y) = x + C ⇒ 1 + y = Ae^x ⇒ y = 4e^x − 1

Apply the Initial Condition:

using y(0) = 1, we substitute x = 0 and y = 1

1 = Ae0 − 1 ⇒ 1 = A − 1 ⇒ A ⇒ 2

Substituting back into the equation gives y = 2e^x −1