Question

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

First order differential equations

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

2020-10-24

Now differentiating this with respect to x
$$dy/dx= d/dx[2-\int_{0}^{x}(1+y(t))\sin tdt]$$
$$dy/dx= d/dx[2]-d/dx[\int_0^x(1+y(t))\sin tdt]$$
$$dy/dx= 0-[(1+y(t))\sin(x) \cdot d/dx(x)-(1+y(0))\sin(0) \cdot d/dx(0)]$$
$$dy/dx= -[(1+y(x))\sin x \cdot (1)-(1+y(0))(0) \cdot 0]$$
$$dy/dx= -[(1+y(x))\sin x-0]$$
$$dy/dx= -(1+y(x))\sin x$$
$$y'= -(1+y)\sin x$$