Question

The general solution of the differential equation \( \frac{dy}{dx} = \frac{y}{x} + \tan \frac{y}{x} \) is

• A. \( \sin \frac{y}{x} = c \)
• B. \( \sin \frac{y}{x} = c x \)
• C. \( \cos \frac{y}{x} = c x \)
• D. \( \cos \frac{y}{x} = c \)

Hint:

When dealing with nonlinear first-order differential equations, substitutions like \( z = \frac{y}{x} \) can often simplify the equation, allowing you to separate the variables and integrate.

Answer 1

The Correct Option is D

Step 1: Understand the differential equation.
The given differential equation is: \[ \frac{dy}{dx} = \frac{y}{x} + \tan \left( \frac{y}{x} \right) \] This is a first-order nonlinear differential equation. To solve it, we will use a method called separation of variables. However, the equation is already suggestive of a transformation that can simplify it. We will introduce a substitution to simplify the equation further.
Step 2: Apply substitution.
Let’s introduce a new variable \( z = \frac{y}{x} \). This substitution simplifies the terms, and we can now express \( y \) as \( y = zx \). Differentiating this expression with respect to \( x \), we get: \[ \frac{dy}{dx} = z + x \frac{dz}{dx} \] Now, substitute this expression for \( \frac{dy}{dx} \) into the original equation: \[ z + x \frac{dz}{dx} = \frac{zx}{x} + \tan z \] Simplify: \[ z + x \frac{dz}{dx} = z + \tan z \]
Step 3: Rearrange and simplify.
Canceling out the \( z \) terms from both sides gives: \[ x \frac{dz}{dx} = \tan z \] Now, separate the variables so that all \( z \)-terms are on one side and all \( x \)-terms are on the other side: \[ \frac{dz}{\tan z} = \frac{dx}{x} \]
Step 4: Integrate both sides.
Now, integrate both sides: \[ \int \frac{dz}{\tan z} = \int \frac{dx}{x} \] The integral on the left side is \( \ln \left| \cos z \right| \), and the integral on the right side is \( \ln |x| \). Thus, we have: \[ \ln \left| \cos z \right| = \ln |x| + C \] Simplifying, we get: \[ \cos z = \frac{1}{x} e^C \] We can replace \( e^C \) with another constant, say \( C' \), and rewrite the equation as: \[ \cos \left( \frac{y}{x} \right) = \frac{C'}{x} \] Thus, the general solution is: \[ \cos \left( \frac{y}{x} \right) = c \] where \( c = C' \) is a constant.
Step 5: Conclusion.
The correct general solution to the differential equation is: \[ \boxed{ \cos \left( \frac{y}{x} \right) = c } \]