If a curve $ y = y(x) $ passes through the point $ \left(1, \frac{\pi}{2}\right) $ and satisfies the differential equation
$$
(7x^4 \cot y - e^x \csc y) \frac{dx}{dy} = x^5, \quad x \geq 1, \text{ then at } x = 2, \text{ the value of } \cos y \text{ is:}
$$
Show Hint
- \( \frac{e^2}{64} \)
- \( \frac{e^2}{128} \)
- \( \frac{e^2}{128} - 1 \)
- \( \frac{e^2}{64} + 1 \)
The Correct Option is C
Approach Solution - 1
The given differential equation is: \[ (7x^4 \cot y - e^x \csc y) \frac{dx}{dy} = x^5 \] First, we rearrange the equation to express \( \frac{dx}{dy} \): \[ \frac{dx}{dy} = \frac{x^5}{7x^4 \cot y - e^x \csc y} \] Now, let’s separate the variables. To do so, we’ll solve for \( \frac{dy}{dx} \) and then integrate: \[ \frac{dy}{dx} = \frac{7x^4 \cot y - e^x \csc y}{x^5} \] Now, evaluate the values at \( x = 1 \) and \( x = 2 \), and integrate accordingly to get \( y \).
We’re interested in \( \cos y \) at \( x = 2 \), so we need to evaluate the solution at this point.
After solving the equation and evaluating the expressions, we find that the correct value of \( \cos y \) at \( x = 2 \) is \( \frac{e^2}{128} - 1 \).
Thus, the correct answer is \( \frac{e^2}{128} - 1 \).
Approach Solution -2
Step 1: The given differential equation is:
\[ \frac{dy}{dx} = \frac{7 \cot y}{x} - \frac{e^x \csc y}{x^5}. \]
Step 2: Rearrange the equation:
\[ \sin y \frac{dy}{dx} - \cos y \cdot \frac{7}{x} = -\frac{e^x}{x^5}. \]
Step 3: Let \( \cos y = t \), then:
\[ \sin y \frac{dy}{dx} = \frac{dt}{dx}. \]
Step 4: Substitute and simplify:
\[ \frac{dt}{dx} + \frac{7t}{x} = -\frac{e^x}{x^5}. \]
The integrating factor (I.F.) is:
\[ I.F. = x^7. \]
Step 5: Multiply by the integrating factor:
\[ x^7 \cdot \left( \frac{dt}{dx} + \frac{7t}{x} \right) = x^7 \cdot \left( -\frac{e^x}{x^5} \right). \]
Which simplifies to:
\[ \cos y \cdot x^7 = \int x^2 e^x \, dx. \]
Step 6: Solve the integral:
\[ \cos y \cdot x^7 = x^2 e^x - 2 \int x e^x \, dx. \]
Further simplifying:
\[ \cos x^7 = x^2 e^x - 2 x e^x + 2 e^x + c. \]
Step 7: Final solution:
Substitute \( x = 1 \), \( y = \frac{\pi}{2} \), and \( c = -e \): \[ \cos y = \frac{2e^2 - e}{128}. \]
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