From the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Solution and Explanation
The equation of the parabola having the vertex at origin and the axis along the positive y-axis is:
x2=4ay...(1)
Differentiating equation (1) with respect to x, we get:
2x=4ay'...(2)
Dividing equation (2) by equation (1) ,we get:
\(\frac{2x}{x^2}=\frac{4ay'}{4ay}\)
\(\Rightarrow \frac{2}{x}=\frac{y'}{y}\)
\(\Rightarrow xy'=2y\)
\(\Rightarrow\) xy'-2y=0
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