Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b:y2= a(b2-x2)
Solution and Explanation
y2= a(b2-x2)
Differentiating both sides with respect to x, we get:
\(2y\frac{dy}{dx}=a(-2x)\)
\(\Rightarrow 2yy'=-2ax\)
\(\Rightarrow yy'=-ax\)...(1)
Again, differentiating both sides with respect to x, we get:
y'.y'+yy''=-a
\(\Rightarrow (y')^2+yy''=-a\)...(2)
Dividing equation(2)by equation(1),we get:
\((y')^2+\frac{yy''}{y'}=-\frac{a}{-ax}\)
\(\Rightarrow xyy''+x(y')^2-yy''=0\)
This is the required differential equation of the given curve.
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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