I come across some problems,(adsbygoogle = window.adsbygoogle || []).push({});

I'm working on finding Differential Equations for n-parameter Family of solutions. However the problem is constructing the n-parameter family equation to work with to find a Differential Equation for the particular n-parameter family equation.

Here's the problem,

Find a differential equation whose solution is

18. A family of circles of fixed radii and centers on the x axis.

Now i know that a family of circles with center at origin is

[tex] x^2 + y^2 = r^2, r > 0 [/tex]

However I do not know how to form an equation such that a family of circles of fixed radii, centers on the x axis. I find that the fixed radii can be represented by a constant?

The answer for 18 is

[tex] (yy')^2 + y^2 = a^2 [/tex]

That is the differential equation, however I do not know how to get the differential equation if I can't construct a family of circles of fixed radii that centers on the x axis.

Here is another problem

19. A family of circles of variable radii, centers on the x axis and passing through the origin.

There is only one problem such as problem 21 that gives the equation to you, so you don't have to construct it.

21. A family of circles with centers in the xy-plane and of variable radii. Then it gives a Hint: Write the equation of the family as

[tex] x^2 + y^2 - 2c_1x - 2c_2y + 2c_3 = 0 [/tex]

I read the problems, and I get an idea of what it wants but, the construction of an equation is difficult.

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# Working on Constructing Equations

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