Hi everyone. I'm stuck with this problem on particular antiderivatives:

"The points (1,3) and (0,2) are on a curve, and at any point (x,y) on the curve, d2y/dx2 = 2 - 4x. Find an equation for the curve."

I was able to get dy/dx from solving, and I got 2x - 2x2 + C. Now I'm unsure what to do... I did, however, try and guess my way through the problem (oops). I eventually came up with the final answer of y = x2 - 2x3/3 - 2x + 14/3. If it happens to actually be the answer, please tell me.

Can someone please lead me in the right direction? Thank you so much!

NOTE: Attached is my solution to the problem just in case you would like to see how I got my (guessed) final answer.

Originally Posted by kaonashi
Hi everyone. I'm stuck with this problem on particular antiderivatives:

"The points (1,3) and (0,2) are on a curve, and at any point (x,y) on the curve, d2y/dx2 = 2 - 4x. Find an equation for the curve."

I was able to get dy/dx from solving, and I got 2x - 2x2 + C. Now I'm unsure what to do... I did, however, try and guess my way through the problem (oops). I eventually came up with the final answer of y = x2 - 2x3/3 - 2x + 14/3. If it happens to actually be the answer, please tell me.

Can someone please lead me in the right direction? Thank you so much!

NOTE: Attached is my solution to the problem just in case you would like to see how I got my (guessed) final answer.
You can check your answer by checking whether the points (1, 3) and (0, 2) satisfy the equation. Clearly (0, 2) doesn't (sub in x=0 and you don't get 2).

You were correct in finding $\displaystyle \frac{dy}{dx} = 2x - 2x^2 + c$

Now do it again to find $\displaystyle y = x^2 -\frac{2}{3}x^3 + cx +d$

Sub in your 2 points, gives two equations with two unknowns, solve for c and d.

No guesswork involved.