It is known that $y(x)$ passes through the points $(0,2)$ and $(1,4)$. Solve for $y(x)$ if the second derivative is:
$$\frac{d^2y}{dx^2} = 1 .$$
The answer is:
$$y = \frac{1}{2}(x^2 + 3x)-2.$$
I've tried integrating both sides of equation… it's probably not the answer. What do I need to do? Is there a technique?
Best Answer
Hint
Assuming that $y$ is a polynomial, since you only have three conditions, it must be a quadratic (since the second derivative is a constant). So $$y=A+B x+C x^2$$ $y''=2C$ implies $C=\frac 12$. Now, compute $A$ and $B$ in order the function goes through the two points.