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Let's analyze each statement in Problem #14 to determine which, if any, satisfy the given initial value problems.

The problem gives three statements with proposed solutions to initial value problems of the form:

1. \( y' = 5x \) with \( y(0) = 3 \)
2. \( y = 5x \), meaning the derivative \( y' = 5 \) (suggested in statement (ii))

The statements are as follows:

(i) \( y = \frac{5x^2}{2} + 3 \):

Which of the following statements are always true?
(i) y = (5x^2)/2 + 3 is a solution to the initial value problem y' = 5x, y(0) = 3.
(ii) y = 3e^(5x^2/2) is a solution to the initial value problem y' = 5x, y(0) = 3.
(iii) y = 3e^(5x) is a solution to the initial value problem y' = 5y, y(0) = 3.

Explore the solution to an initial value problem in differential equations. Determine if statements (i), (ii), and (iii) satisfy the given initial conditions for differential equations y' = 5x, y(0) = 3 and y' = 5y, y(0) = 3. Understand exponential solutions and algebraic manipulation techniques.

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