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

To solve the equation, we first simplify it by dividing both sides by \( \cos\left(\frac{y}{x}\right) \), given \( \cos\left(\frac{y}{x}\right) eq 0 \): \[ x \frac{dy}{dx} = y + \frac{x}{\cos\left(\frac{y}{x}\right)} \] The next step is to separate the variables. Recognizing that the equation involves \( x \) and \( y \) within a complex trigonometric function, we employ the substitution \( v = \frac{y}{x} \), which implies \( y = vx \). Differentiating \( y = vx \) with respect to \( x \) yields: \[ \frac{dy}{dx} = v + x \frac{dv}{dx} \] Substituting this into the original equation: \[ x \left(v + x \frac{dv}{dx}\right) = y + \frac{x}{\cos(v)} \] Replacing \( y \) with \( vx \): \[ x \left(v + x \frac{dv}{dx}\right) = vx + \frac{x}{\cos(v)} \] Simplifying both sides: \[ xv + x^2 \frac{dv}{dx} = vx + \frac{x}{\cos(v)} \] The \( vx \) terms on both sides cancel out: \[ x^2 \frac{dv}{dx} = \frac{x}{\cos(v)} \] Dividing by \( x \) further simplifies the equation: \[ x \frac{dv}{dx} = \frac{1}{\cos(v)} \] Now, we separate the variables: \[ \cos(v) dv = \frac{dx}{x} \] Integrating both sides: \[ \int \cos(v) dv = \int \frac{dx}{x} \] The integration results in \( \sin(v) \) for the left side and \( \ln|x| \) for the right side: \[ \sin(v) = \ln|x| + C \] Finally, we substitute back \( v = \frac{y}{x} \) to obtain the solution in terms of \( x \) and \( y \): \[ \sin\left(\frac{y}{x}\right) = \ln|x| + C \] This represents the general solution to the given differential equation: \[ \boxed{\sin\left(\frac{y}{x}\right) = \ln|x| + C} \]