Question

For each of the differential equation, find the general solution: y log y dx - x dy = 0

Answer 1

The general solution is y=e^cx.

Hope it helps..:)

Answer 2

Answer:

$y\ =\ e^{cx}$ where C is the arbitrary constant.

Step-by-step explanation:

In this question,

We need to find the general solution of y log y dx - x dy = 0

According to the question,

y log y dx - x dy = 0

y log y dx = x dy

$\frac{dy}{y log y}\ =\ \frac{dx}{x}$

Integrating both the sides we get,

$\int{\frac{dy}{y log y}}\ =\ \int{\frac{dx}{x}}$

Using substitution

Let log y = t

Differentiating with respect to y we get,

$\frac{1}{y}\ =\ \frac{dt}{dy}$

dy = y dt

On Substituting we get

$\int{\frac{dt}{t}}\ =\ \int{\frac{dx}{x}}$

log|t| = log|x| + logC                {Where C is the arbitrary constant}

Replacing t = log y we get,

log(log(y)) = log Cx                      {log a + log b = log ab}

log y = Cx

Therefore,$\boxed{\mathbf{y\ =\ e^{Cx}}}$