## Wednesday, April 9, 2014

### Finding Equation - Curve, 13

Category: Differential Equations, Integral Calculus, Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the equation of the curve for which y" = 6x², and which passes through the points (0, 2) and (-1, 3).

Solution:

The concavity of a curve is equal to the second derivative of a curve with respect to x. In this case, y" = d²y/ dx². Let's consider the given concavity of a curve

We can rewrite the above equation as follows

Multiply both sides of the equation by dx, we have

Integrate on both sides of the equation, we have

Since the slope of a curve is not given in the problem but the two points are given, then we have to continue the integration until we get an equation in terms of x and y. Let's consider the equation above

Multiply both sides of the equation by dx, we have

Integrate on both sides of the equation, we have

In order to get the value of arbitrary constants, we need to use the coordinates of two points so that we can form the two equations, two unknowns.

By using the point (0, 2), substitute the value of x and y to the above equation, we have

By using the point (-1, 3), substitute the value of x and y to the above equation, we have

but

then the above equation becomes

Therefore, the equation of a curve is