High School

A first order linear differential equation is given by:
dy/dx=-k(y-30)
Using integrating factor method, find the general solution for the differential equation.
Given y(0)=90 and y(1)=38.1

Answer :

The general solution to the first order linear differential equation dy/dx = -k (y-30) using the integrating factor method is y = 30 + 60e^(-2x).

First, rewrite the equation in the standard form: dy/dx + ky = k * 30.

Here, P(x) = k and Q(x) = 30k.

To find the solution using the integrating factor method:

  1. Identify the integrating factor, which is given by IF = e^(∫P(x) dx) = e^(∫k dx) = e^(kx).
  2. Multiply the entire differential equation by the integrating factor: e^(kx) * dy/dx + e^(kx) * k * y = e^(kx) * 30k.
  3. Notice that the left-hand side is the derivative of (y * e^(kx)): d/dx (y * e^(kx)) = e^(kx) * 30k.
  4. Integrate both sides with respect to x: ∫ d/dx (y * e^(kx)) dx = ∫ 30k * e^(kx) dx.
  5. This simplifies to: y * e^(kx) = 30 * e^(kx) + C, where C is the constant of integration.
  6. Solving for y, we get: y = 30 + Ce^(-kx).

Next, use the initial conditions to find C and k:

  1. Using y(0) = 90: 90 = 30 + C * e^(0)
  2. Therefore, C = 60.
  3. Using y(1) = 38.1: 38.1 = 30 + 60e^(-k)
  4. Therefore, 8.1 = 60e^(-k), which gives e^(-k) = 0.135 and hence k ≈ 2.

Thus, the general solution is y = 30 + 60e^(-2x).

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