Mathematics

Problem

Let

G

be the function defined for all real numbers by

G(x)=2+\int_{1}^{x}\left(3t^{2}-4t+1\right)\,dt.

It is proposed that

G

satisfies the differential equation

\dfrac{dG}{dx}=3x^{2}-4x+1

. (a) Evaluate the definite integral and write

G(x)

in simplest polynomial form. (b) Verify directly from your result in part (a) that

G

satisfies the differential equation

\dfrac{dG}{dx}=3x^{2}-4x+1

for every real

x

. (c) Explain why

G

is the unique solution of the differential equation that passes through the point

(1,2)