Question 94 of 96beginner🔧 ApplyMCQ1 mark

The general solution of the differential equation ydx−xdyy=0\frac{y dx - x dy}{y} = 0 is

Exercise: Miscellaneous Exercise on Chapter 9 | Q: 13 | (Page 36)
For More Understanding

Explanation

Option (C) is the correct answer. The differential equation provided can be solved using the method of separation of variables, as demonstrated in Example 5 and Example 6 of the provided context. By algebraically rearranging the equation to group all terms containing 'x' on one side and all terms containing 'y' on the other, we can integrate both sides to find the general solution.

Solution Steps

  1. Step 1: The given differential equation is ydx−xdyy=0\frac{y dx - x dy}{y} = 0. We simplify this to ydx−xdy=0y dx - x dy = 0.

  2. Step 2: Rearranging the terms to separate variables, we get ydx=xdyy dx = x dy, which further simplifies to dxx=dyy\frac{dx}{x} = \frac{dy}{y}.

  3. Step 3: Integrating both sides (as shown in Example 5), we write ∫dxx=∫dyy\int \frac{dx}{x} = \int \frac{dy}{y}.

  4. Step 4: Solving the integrals, we get log⁡x=log⁡y+log⁡C\log x = \log y + \log C, which can be rearranged as log⁡x−log⁡y=log⁡C\log x - \log y = \log C or log⁡(xy)=log⁡C\log(\frac{x}{y}) = \log C.

  5. Step 5: Taking the antilogarithm, we find xy=C\frac{x}{y} = C, which implies y=1Cxy = \frac{1}{C}x. Replacing the constant 1C\frac{1}{C} with a new constant CC, the general solution is y=Cxy = Cx. This matches Option (C).

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