Obtain the differential equation of the family of

Kiel Cruz

Kiel Cruz

Answered question

2022-09-26

Obtain the differential equation of the family of curves described.

1.      Straight lines with slope equal to twice of the x-intercept

2.      Circles with fixed radius r and tangent to y-axis

3.      Parabolas with vertex and focus on the y-axis

Answer & Explanation

Eliza Beth13

Eliza Beth13

Skilled2023-05-31Added 130 answers

To obtain the differential equations of the given family of curves, let's consider each case separately:
1. Straight lines with slope equal to twice the x-intercept:
Let's assume the equation of a straight line in slope-intercept form as y=mx+c, where m is the slope and c is the y-intercept.
Given that the slope is twice the x-intercept, we have m=2c. Substituting this into the equation, we get y=2cx+c.
To find the differential equation, we need to express y as a function of x and its derivative. Here, we have dydx=2c. Rearranging the terms, we have c=12dydx.
Substituting this value of c into the equation, we get y=xdydx+12dydx.
Therefore, the differential equation for the family of straight lines with a slope equal to twice the x-intercept is y=xdydx+12dydx.
2. Circles with fixed radius r and tangent to the y-axis:
The equation of a circle with radius r and center (h,k) is given by (xh)2+(yk)2=r2.
Here, the circle is tangent to the y-axis, so its center lies on the y-axis. Therefore, the equation of the circle becomes x2+(yk)2=r2.
To find the differential equation, we need to express y as a function of x and its derivative. Differentiating both sides of the equation with respect to x, we get 2x+2(yk)dydx=0.
Simplifying, we have x+(yk)dydx=0.
Therefore, the differential equation for the family of circles with a fixed radius r and tangent to the y-axis is x+(yk)dydx=0.
3. Parabolas with vertex and focus on the y-axis:
The equation of a parabola with vertex at the origin is given by y2=4ax, where a is the distance from the vertex to the focus.
Here, the vertex and focus are both on the y-axis, so a is the distance from the origin to the focus.
To find the differential equation, we need to express y as a function of x and its derivative. Differentiating both sides of the equation with respect to x, we get 2ydydx=4a.
Simplifying, we have ydydx=2a.
Therefore, the differential equation for the family of parabolas with the vertex and focus on the y-axis is ydydx=2a.
These are the differential equations for the given families of curves.

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