The differential equation has general solution

MATHEMATICS · Single correct · A Level · Mathematics · Differentiation · Differentiation

Problem

The differential equation

\dfrac{dy}{dx}=y

has general solution

y=Ce^{x}

. Which of the following best explains why this form indicates the equation possesses infinitely many solutions? (a) Because the derivative of

Ce^{x}

equals

Ce^{x}

, the function repeats itself indefinitely. (b) Because adding any constant to

e^{x}

still produces a solution curve, giving infinitely many vertical shifts. (c) Because the constant

C

can assume any real value, producing a distinct function for each choice. (d) Because the exponential function is unbounded, any initial value of

y

can be reached.