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Given, A differential equation

To find: Find the solution of the differential equation , for y(1) = - 1

Explanation: We have given a differential equation

We can write it as ,

Now, integrating both sides,

Here, we know, and

For y, and

So,

…(i)

Now, y(1) = - 1 , then

Put x = 1 and y = - 1 in equation (i)

- 1 - 2 log 1 = 1 + 2 log 1 + C

- 1 = 1 + C

C = - 2

On putting C = - 2 in equation (i), we get

Since, mlog n = log n^m

y - x + 2 = log (y + 2)² + log x²

Since, log m + log n = log mn

y - x + 2 = log (x²(y + 2)²)

Hence, The solution of the given differential equation is

y - x + 2 = log(x²(y + 2)²)