Question

Solve the differential equation: \[ x \cos\left(\frac{y}{x}\right) \frac{dy}{dx} = y \cos\left(\frac{y}{x}\right) + x \]

Hint:

To solve complex differential equations, try substitution to simplify the relationship between variables. In this case, we used \( v = \frac{y}{x} \).

Answer 1

We start by simplifying the equation. Divide both sides by \( \cos\left(\frac{y}{x}\right) \) (assuming \( \cos\left(\frac{y}{x}\right) \neq 0 \)): \[ x \frac{dy}{dx} = y + \frac{x}{\cos\left(\frac{y}{x}\right)} \] Now we aim to separate the variables. Observe that the equation involves both \( x \) and \( y \) in terms of a complicated trigonometric function. To proceed, we make the substitution \( v = \frac{y}{x} \), so that \( y = vx \). Now, differentiate \( y = vx \) with respect to \( x \): \[ \frac{dy}{dx} = v + x \frac{dv}{dx} \] Substitute this expression for \( \frac{dy}{dx} \) into the original equation: \[ x \left(v + x \frac{dv}{dx}\right) = y + \frac{x}{\cos(v)} \] Since \( y = vx \), we substitute for \( y \): \[ x \left(v + x \frac{dv}{dx}\right) = vx + \frac{x}{\cos(v)} \] Simplify both sides: \[ xv + x^2 \frac{dv}{dx} = vx + \frac{x}{\cos(v)} \] Cancel out the \( vx \) terms on both sides: \[ x^2 \frac{dv}{dx} = \frac{x}{\cos(v)} \] Divide both sides by \( x \) to simplify further: \[ x \frac{dv}{dx} = \frac{1}{\cos(v)} \] Now, separate the variables: \[ \cos(v) dv = \frac{dx}{x} \] Integrate both sides: \[ \int \cos(v) dv = \int \frac{dx}{x} \] The integral of \( \cos(v) \) is \( \sin(v) \), and the integral of \( \frac{1}{x} \) is \( \ln|x| \): \[ \sin(v) = \ln|x| + C \] Substitute back \( v = \frac{y}{x} \) to express the solution in terms of \( x \) and \( y \): \[ \sin\left(\frac{y}{x}\right) = \ln|x| + C \] Thus, the general solution to the differential equation is: \[ \boxed{\sin\left(\frac{y}{x}\right) = \ln|x| + C} \]