From the differential equation of the family of circles touching the y-axis at the origin.
From the differential equation of the family of circles touching the y-axis at the origin.
Solution and Explanation
The Centre of the circle touching the y-axis at origin lies on the x-axis.
Let (a,0) be the Centre of the circle.
Since, it touches the y-axis at origin, its radius is a.
Now, the equation of the circle with Centre (a,0)and radius a is
\((x-a)^2+y^2=a^2\).
\(\Rightarrow x^2+y^2=2ax\)...(1)
Differentiating equation (1)with respect to x, we get:
2x+2yy'=2\(a\)
\(\Rightarrow\) x+yy'= \(a\)
Now, on substituting the value of a in equation(1),we get:
x2+y2=2(x+yy')x
\(\Rightarrow\) x2+y2=2x2+2xyy'
\(\Rightarrow\) 2xyy'+x2=y2
This is the required differential equation.
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Concepts Used:
General Solutions to Differential Equations
A relation between involved variables, which satisfy the given differential equation is called its solution. The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution and the solution free from arbitrary constants is called particular solution.
For example,
Read More: Formation of a Differential Equation