Alan, Betty and Catherine had to find a solution of the differential equation \frac{d y}{d x}=3 x^{2}(y-4) After much discussion, \text { Alan said: } y(x)=2 e^{x}+3 \text {, } \text { Betty said: } y(x)=e^{x^{3}}+4, \text { and } \text { Catherine said: } v(x)=4 \text {. } (a) Who is right? Justify your answer by taking derivatives. (b) Which of the functions suggested by Alan, Betty and Catherine satisfies the differential equation and the initial condition: y(0) = 5?

Fig: 1

Fig: 2

Fig: 3

Fig: 4

Fig: 5

Fig: 6

Fig: 7

Fig: 8