Question

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

(a) \(cot(\frac{y}{x}) = xc\)

(b) \(cos(\frac{y}{x}) = xc\)

(c) \(sec^2(\frac{y}{x}) = xc\)

(d) \(sin(\frac{y}{x}) = xc\)

This question was posed to me by my college professor while I was bunking the class.

Asked question is from Separable and Homogeneous Equations in section Ordinary Differential Equations – First Order & First Degree of Engineering Mathematics

Answer 1

The correct choice is (d) \(sin(\frac{y}{x}) = xc\)

To explain: \(\frac{dy}{dx} = \frac{y}{x} + tan⁡\frac{y}{x}\) we can clearly see that it is an homogeneous equation

substituting \(y = vx \rightarrow \frac{dy}{dx} = v + x \frac{dv}{dx} = v + tan \,v\)

separating the variables and integrating we get

\(\int \frac{1}{tan⁡v} \,dv = \int \frac{1}{x} dx\)

log(sin v) = log x + log c

\(sin \,v = xc \rightarrow sin(\frac{y}{x}) = xc\) is the solution where c is constant.