Question

The general solution of \( \frac{dy}{dx} = \cos^2(x - y - 1) \) is given by \( x = \):

• A. \( C - \cot(x - y - 1) \)
• B. \( C - \tan(x - y + 1) \)
• C. \( y + C \cot(x - y - 1) \)
• D. \( Cy + \tan(x - y - 1) \)

Hint:

Substitution can simplify differential equations with repeating terms.

Answer 1

The Correct Option is A

Step 1: Substitution
Let $v = x - y - 1$. Then $\dfrac{dv}{dx} = 1 - \dfrac{dy}{dx}$.


Step 2: Substitute $\dfrac{dy}{dx}$
Given $\dfrac{dy}{dx} = \cos^2(v)$, so:
$\dfrac{dv}{dx} = 1 - \cos^2(v) = \sin^2(v)$.


Step 3: Separate and integrate
$\dfrac{dv}{\sin^2(v)} = dx \Rightarrow \int \csc^2(v) \, dv = \int dx$
$- \cot(v) = x + C_1$


Step 4: Substitute back $v$
$- \cot(x - y - 1) = x + C_1$


Step 5: Solve for $x$
$x = - \cot(x - y - 1) - C_1$
Let $C = -C_1$, then:
$x = C - \cot(x - y - 1)$