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This is what I did, but not sure what to do next:

cos(y) dy = -0.5 dx

Integral \[cos(y) dy\] = Integral \[-0.5 dx\]

sin y = -0.5 dx + C

How do I solve this and get rid of sin? I did arcsin of both sides but the answer seemed to be wrong

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You're on the right path. Integral of cosy would be siny. Integral of -0.5dx would be -0.5x + c. Now, using the initial condition, find the value of c and substitute it into the equation. That'll be the final needed differential equation.

Edit: In general, once you isolate the variables like you did, just go ahead and integrate.
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Integrating cos(y) with respect to y is the same as integrating cos(x) with respect to x, the answer is just sin(y). The other side becomes -0.5x+c, and you can solve for c by substituting x=0 and y=pi/4 into the resulting equation.

Then use inverse sin on both sides to get the final answer of y as a function of x
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