Question

Find the general solution of the differential equation \(x\cos\left(\dfrac yx\right)\dfrac{dy}{dx} = y\cos\left(\dfrac yx\right)+x\).

Hint:

Homogeneous equation: substitute y=vx, dy/dx=v+x dv/dx, then separate variables.

Answer 1

Step 1: Understanding the Concept:
The right side, divided by \(x\), depends only on \(y/x\): \(\dfrac{dy}{dx} = \dfrac{y}{x}+\dfrac{1}{\cos(y/x)}\), which is a function of \(y/x\) alone. This makes the equation homogeneous, solvable with the substitution \(y=vx\).

Step 2: Substituting \(y=vx\):
With \(y=vx\), \(\dfrac{dy}{dx} = v+x\dfrac{dv}{dx}\). The original equation becomes:
\[ x\cos(v)\left(v+x\frac{dv}{dx}\right) = vx\cos(v)+x \]

Step 3: Expanding and simplifying:
\[ xv\cos(v) + x^2\cos(v)\frac{dv}{dx} = vx\cos(v)+x \]
The \(xv\cos v\) terms on each side cancel:
\[ x^2\cos(v)\frac{dv}{dx} = x \implies \cos(v)\frac{dv}{dx} = \frac1x \]

Step 4: Separating variables and integrating:
\[ \cos(v)\,dv = \frac{dx}{x} \]
\[ \int\cos v\,dv = \int\frac{dx}{x} \]
\[ \sin v = \ln|x| + C \]

Step 5: Substituting back \(v=y/x\):
\[ \sin\left(\frac{y}{x}\right) = \ln|x| + C \]

Final Answer:
The general solution is \(\sin(y/x) = \ln|x|+C\). \[ \boxed{\sin\left(\dfrac{y}{x}\right)=\ln|x|+C} \]