# Write the first order differential equation for y=2-int_0^x(1+y(t))sin tdt

Write the first order differential equation for $y=2-{\int }_{0}^{x}\left(1+y\left(t\right)\right)\mathrm{sin}tdt$

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brawnyN

Now differentiating this with respect to x
$dy/dx=d/dx\left[2-{\int }_{0}^{x}\left(1+y\left(t\right)\right)\mathrm{sin}tdt\right]$
$dy/dx=d/dx\left[2\right]-d/dx\left[{\int }_{0}^{x}\left(1+y\left(t\right)\right)\mathrm{sin}tdt\right]$
$dy/dx=0-\left[\left(1+y\left(t\right)\right)\mathrm{sin}\left(x\right)\cdot d/dx\left(x\right)-\left(1+y\left(0\right)\right)\mathrm{sin}\left(0\right)\cdot d/dx\left(0\right)\right]$
$dy/dx=-\left[\left(1+y\left(x\right)\right)\mathrm{sin}x\cdot \left(1\right)-\left(1+y\left(0\right)\right)\left(0\right)\cdot 0\right]$
$dy/dx=-\left[\left(1+y\left(x\right)\right)\mathrm{sin}x-0\right]$
$dy/dx=-\left(1+y\left(x\right)\right)\mathrm{sin}x$
${y}^{\prime }=-\left(1+y\right)\mathrm{sin}x$