If y=y(x) is the solution of differential equation ysinxdydx=cosx(sinx−y2) where x≠nπ,n∈I and y(π2)=√23. Then 9y4(π3)=
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Solution
Given : ysinxdydx=cosx(sinx−y2) ⇒ydydx+y2cotx=cosx
put y2=z,2ydydx=dzdx ⇒dzdx+z(2cotx)=2cosx (linear form) I.F.=e∫2cotxdx=sin2x
solution is given by, z(sin2x)=∫(2cosx⋅sin2x)dx⇒y2(sin2x)=2sin3x3+C
As, y(π2)=√23,C=0 ⇒y2=2sinx3∴9y4(π3)=3