Obtain the Differential equations: parabolas with vertex and focus on the x-axis.

Question
Differential equations
asked 2020-11-10
Obtain the Differential equations: parabolas with vertex and focus on the x-axis.

Answers (1)

2020-11-11
Let M denote the family of parabolas whose vertex and focus both on the x-axis and let (a,0) be the focus of a member of the given family, where a is an arbitrary constant. Therefore, equation of family M is \(\displaystyle{y}^{{2}}={4}{a}{x}\)...(1) Differentiating both sides of equation with respect to x, we get
\(\displaystyle{2}{y}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={4}{a}{x}\)
Substituting the value of 4a from equation in (1) \(\displaystyle{y}^{{2}}={\left({2}{y}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}\right)}{x}\)
\(\displaystyle\to{y}^{{2}}={2}{x}{y}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}\)
\(\displaystyle\to{y}^{{2}}-{2}{x}{y}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}=-\)
which is the Differential equations of the family of parabolas.
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