What is the general solution of the differential equation : \(\frac{y

1.	What is the general solution of the differential equation : \(\frac{ydx - xdy}{y} = 0\)?
Answer» Given differential equation is \(\frac{ydx - xdy}{y} = 0\) ⇒ dx - \(\frac{x}{y}\)dy = 0 ⇒ dx = \(\frac{x}{y}\)dy ⇒ \(\frac{1}{x}\)dx = \(\frac{1}{y}\)dy \(\int \) \(\frac{1}{x}\)dx = \(\frac{1}{y}\)dy (By integrating both sides of above equation) ⇒ log x = log y + log C, where log is an integral constant. ⇒ log x = log Cy ( \(\because\) log a + log b = log ab) ⇒ x = Cy, which is general solution of given differential equation. (By taking anti-log both sides of above equation. ) Hence, the general solution of the differential equation \(\frac{y dx - x dx}{y} = 0\)is x = Cy.

What is the general solution of the differential equation : \(\frac{ydx - xdy}{y} = 0\)?

Answer»

Given differential equation is \(\frac{ydx - xdy}{y} = 0\)

⇒ dx - \(\frac{x}{y}\)dy = 0

⇒ dx = \(\frac{x}{y}\)dy ⇒ \(\frac{1}{x}\)dx = \(\frac{1}{y}\)dy

\(\int \) \(\frac{1}{x}\)dx = \(\frac{1}{y}\)dy (By integrating both sides of above equation)

⇒ log x = log y + log C, where log is an integral constant.

⇒ log x = log Cy ( \(\because\) log a + log b = log ab)

⇒ x = Cy, which is general solution of given differential equation.

(By taking anti-log both sides of above equation. )

Hence, the general solution of the differential equation \(\frac{y dx - x dx}{y} = 0\)is x = Cy.

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