Question

Can I know the steps​

Answer 1

Step-by-step explanation:

We are given:

$xy'=coty$

This can be rewritten as

$x\frac{dy}{dx} =coty$

Next, we can bring the x's and y's to their respective sides by dividing by coty and x and then multiplying the dx to the other side. We can then change $\frac{1}{coty}$ into $tany$. This gives us the differential

$tany\,dy=\frac{1}{x} \,dx$

Now we can integrate each side

$\int tan(y)\,dy=\int \frac{1}{x} \,dx$

To integrate tan(y), we need to manipulate it

$\int \frac{sin(y)}{cos(y)} \,dy=\int \frac{1}{x} \,dx$

Now we can use u-substitution where $u= cos(y)\\du=-sin(y) dy$

This gives us

$-\int \frac{1}{u} \,du=\int \frac{1}{x} \,dx$

Now, lets integrate both sides

$-ln|u|=ln|x|+c$

Next, we can substitute our u value back in

$-ln|cos(y)|=ln|x|+c$

Now we can add $-ln|cos(y)|$ to the other side and subtract c from each side. This gives us

$C_2=ln|x|+ln|cos(y)|$

Next, we can apply a property of logarithms to combine this sum of two logs into one log.

$C_2=ln|xcos(y)|$

Lastly, we can add a base e to each side to remove the ln

$C_3=|xcos(y)|$

And here is our answer.