Specific antiderivatives

Normally the antiderivative is in form of f(x) +C. But actually we could use some additional information to get the C and get the function only in terms of x. And we often call the "additional information" as Initial Conditions, or f₀(x).

Example

Solve:

• We could Integrate the f'(x) to get f(x) = 9eˣ + C.

• Since f(8) = 9e⁸ - 8, we could easily see that C = -8

• So the function under this condition would be: f(x) = 9eˣ -8

• Then we will get the result f(0) = 9*1 - 8 = 1.

Example

Solve:

• We could integrate f'(x) to get f(x) = x³ - x² + 7x +C

• Since f(6)=200, so we could substitute 6 into f(x):

• f(6) = 6³ - 6² + 7*6 +C = 200 which results C = -22

• So the function under this condition should be f(x) = x³ - x² + 7x - 22

• And f(1) = 1³ - 1² + 7*1 - 22 = -15

Example (Separable equations with specific solutions)

Solve:

• It looks confusing, but let's assume f(x) as y so dy/dx = 2y

• Separate differential equations to get dy/y = 2dx

• Take integral of both sides:

  

• Since f(1) = 5, so:

  

• And f(3) = 5e⁴, which means m = 5, n = 4

Example

Solve: Hint: f(0) = 2

Example

Solve Hint: Don't need to solve y completely.