Answer to Question #280822 in Differential Equations for Shell

Question #280822

Find the equation of the curve at every point that passes through the point (0, −1) and which the normal line at any point (x, y) has a slope of 1/(3x + 2).

1
Expert's answer
2021-12-21T17:01:18-0500

Let the equation of the curve be y=f(x).y=f(x). Then slope of the nornal to the curve at any point (x,y)(x, y) will be

slope=m=1f(x)slope=m=-\dfrac{1}{f'(x)}

Given slope=m=13x+2.slope=m=\dfrac{1}{3x+2}.

Then


f(x)=1m=(3x+2)f'(x)=-\dfrac{1}{m}=-(3x+2)

Integrate


f(x)=(3x+2)dx=32x22x+Cf(x)=-\int(3x+2)dx=-\dfrac{3}{2}x^2-2x+C

The curve passes through the point (0,1)(0, −1)


1=32(0)22(0)+C=>C=1-1=-\dfrac{3}{2}(0)^2-2(0)+C=>C=-1

The equation of the curve at every point is


y=32x22x1y=-\dfrac{3}{2}x^2-2x-1


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