The given differential equation can be written as
dxdy+ysec2x=tanx.sec2x
Integrating factor =e∫sec2xdx=etanx
Hence, solution of given differential equation,
y.etanx=∫tanx.sec2x.etanxdx
⇒y.etanx=tanx.etanx−∫sec2x.etanxdx (using integration by parts)
⇒y.etanx=tanx.etanx−etanx+c
Given, y(0)=0⇒c=1
Solution of given differential is
y.etanx=tanx.etanx−etanx+1
Hence, y(−4π)=e−1(−1)e−1−e−1+1
⇒y(−4π)=e−2