Answer to Question #306114 in Differential Equations for Rach

Question

the curve y = f(x) has a y-intercept of 3. the tangent to this curve at the point (x, y) has an x-intercept of x+2. What is the equation of this curve?

Answer 1

The slope of the tangent to the curve at the point $(x, y)$  is

$$y'=x+2$$

Integrate

$$y=\dfrac{1}{2}x^2+2x+C$$

The curve $y = f(x)$ has a y-intercept of 3

$$3=0+0+C$$

$$C=3$$

The equation of this curve is

$$y=f(x)=\dfrac{1}{2}x^2+2x+3$$