Find the general solution of the differential equation specified.

1) dy/dt= 1/(ty+1+y+1)

2) dy/dx=sec y

I got y(x)=arcsin(x) for the second one. I'm not sure what to do with the first one.

To find the general solution of a differential equation, we need to integrate both sides of the equation with respect to the respective variables and then solve for the dependent variable.

Let's solve each differential equation one by one:

1) dy/dt = 1/(ty + 1 + y + 1)

To solve this, we can rewrite the equation as follows:

dy / (ty + y + 2) = dt

Now, we need to separate the variables y and t. To do this, we can use partial fraction decomposition or a substitution.

Let's try the substitution method:

Let u = ty + y + 2
Then du = (t + 1)dy

Substituting back into the equation:

du / u = dt / (t + 1)

Integrating both sides:

ln|u| = ln|t + 1| + C

where C is the constant of integration.

Now, substituting back for u:

ln|ty + y + 2| = ln|t + 1| + C

Using the properties of logarithms:

ty + y + 2 = (t + 1)e^C

Rearranging the terms:

ty + y = (t + 1)e^C - 2

Now, we need to solve for y in terms of t. This may not have an elementary solution, but we can still express the general solution implicitly.

2) dy/dx = sec(y)

To solve this, we can separate the variables and integrate both sides:

sec(y) dy = dx

Integrating both sides:

∫sec(y) dy = ∫dx

The integral of sec(y) is ln|sec(y) + tan(y)|, and the integral of dx is x:

ln|sec(y) + tan(y)| = x + C

Now, we can solve for y:

sec(y) + tan(y) = e^(x + C)

This gives us an implicit solution for the differential equation.

For the second equation, you mentioned that you already got y(x) = arcsin(x). However, the differential equation you provided (dy/dx = sec(y)) does not match the solution you stated. Therefore, please double-check your calculations or provide additional information if needed.