Question

find the general solution of the following:

1. dy/dt=t/(ye^(y+t^2))
2. dy/dx=cosxe^(y+sinx)
3. dy/dx=2x*sqrt(1-y^2)

Answer 1

Each of these equations are separable:
\[\begin{array}{c}
\dfrac{dy}{dt}=\dfrac{t}{e^{y+t^2}}~~\iff~~e^y\,dy=te^{-t^2}\,dt\\\\
\hline\\
\dfrac{dy}{dx}=\cos x~e^{y+\sin x}~~\iff~~
e^{-y}\,dy=\cos x~e^{\sin x}\,dx\\\\
\hline\\
\dfrac{dy}{dx}=2x\sqrt{1-y^2}~~\iff~~
\dfrac{dy}{\sqrt{1-y^2}}=2x\,dx
\end{array}\]

Answer 2

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