Find an equation of the form y=f(x) for a curve with x-intercept 1 whose tangent line at any point (x,y) has slope x^(3)e^(-y).

Question

Find an equation of the form y=f(x) for a curve with x-intercept 1 whose tangent line at any point (x,y) has slope x^(3)e^(-y). 

Answer 1

We need two things here:

An x-intercept of 1 means that f(1) = 0

Secondly, realize that the derivative is the slope of a tangent line at any point (x, y).

Thus, the second part of the problem says dy/dx = x³e^-y

You can solve for y because this differential equation is separable.

Do you think, you'd be able to solve it now?