Question

In problems 9-12, a family of solutions of a differential equation is given. Find the value of the constant C so the solution satisfies the initial value condition.

9. y' = 2x and y(3) 7. y=x² + C. 10. y- = 3x - 5 and y(1) = 2. y=x³-5x+C
11. y' = 3y² and y(0) = 5. y=Ce^3x
12. y' = -2y and y(0)=3. y = Ce^=2x

Answer 1

9. y' = 2x and y(3) 7. y=x2 + C.

Plugging in x=3 and y=7 into the equation y= x^2 + C, we get:

7 = 9 + C

C = -2

10. y' = 3x - 5 and y(1) = 2. y = x^3 - 5x + C

Plugging in x=1 and y=2 into the equation y = x^3 - 5x + C, we get:

2 = 1 - 5 + C

C = 6

11. y' = 3y^2 and y(0) = 5. y = Ce^3x

Plugging in x=0 and y=5 into the equation y = Ce^3x, we get:

5 = C(e^0) = C

C = 5

12. y' = -2y and y(0) = 3. y = Ce^(-2x)

Plugging in x=0 and y=3 into the equation y = Ce^(-2x), we get:

3 = C

C = 3