In the following problems, obtain the differential equations of the given families of curves whose properties are indicated Parabolas with vertex on the y-axis, with axis parallel to the x-axis, and with distance from the focus to vertex fixed at a.

Name: In the following problems, obtain the differential equations of the given families of curves whose properties are indicated Parabolas with vertex on the y-axis, with axis parallel to the x-axis, and with distance from the focus to vertex fixed at a.
Uploaded: 2022-02-23T17:22:57+00:00
Description: In the following problems, obtain the differential equations of the given families of curves whose properties are indicated Parabolas with vertex on the y-axis, with axis parallel to the x-axis, and with distance from the focus to vertex fixed at a.

Question

In the following problems, obtain the differential equations of the given families of curves whose properties are indicated   Parabolas with vertex on the y-axis, with axis parallel to the x-axis, and with distance from the focus to vertex fixed at a.

Esta Hurtado · Accepted Answer

The parabola will be of type Y2=4AX, thus the equation of the parabola is(y−h)2=4a(x−0) where a is the distance between focus and vertex.Differentiate (y−h)2=4a(x−0) with respect to x to obtain equation 1(y−h)2=4a(x−0)2(y−h)dydx=4a2(y−h)y′=4a (1)Differentiate 2(y−h)y′=4a with  respect to x to obtain equation 2.2(y−h)y′=4a2(y−h)y″+2(y′)2=0y″(y−h)+(y′)2=0 (2)Solve equation 2 and obtain the value of h in terms of y&#039; and y&#039;&#039;.y″(y−h)+(y′)2=0yy″−hy″+(y′)2=0−hy″=−(y′)2−yy″−h=−((y′)2+yy″)y″h=((y′)2+yy″)y″Substitute the value of h in equation 1 and obtain the value of a.2(y−h)y′=4a2(y−(((y′)2+yy″}{y″}y′=4a2(yy″−(y′)2+yy″y″)y′=4a2(2yy″−(y′)2y″)y′=4a2y(y′)y″−(y′)3

intacte87 · Accepted Answer

The equation of the circle will be of the formx2+y2=16The differential equation in this case will be2x+2yy′=02yy′=−2xdydx=−xyThe parabola with the vertex on the y-axis will be of the form(y−h)2=4a(x−k)Hence the equation in this case will be of the formy′=2ayNote - Post any doubts/queries in comments section.

alenahelenash · Accepted Answer

Two way parabola possible for given questionEquation of parabolaY2=4AX(y−h)2=4a(x−0)(y−h)2=4ax−i)Differentiate equation i)2(y−h)dy/dx=4a2(y−h)y′=4a−ii)Again differentiate2y′(y−h)+(y′)2=0Y″y−y″h+y′2=0h=[y+y2/y′]put value of h in ii)2(y−y−y2/y′)y′=4a[−1/2y2y′/y″=a]Put values of a and h in eq i)(y−y−y2/y′)2=−2⋅y2y′x/y″y2=−2⋅y′y″xy2=−2dy/dxd2y/dx2xy2+2dy/dxd2y/dx2x=0

In the following problems, obtain the differential equations of the

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