Solve the differential equation: ysinxdydx=cosx(sinx−y2)
A
y2=23sinx+csin2x
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B
y2=−23sinx+csin2x
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C
y2=23sinx−csin2x
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D
y2=23cosx+csin2x
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Solution
The correct options are Ay2=23sinx+csin2x Cy2=23sinx−csin2x Given differential eqn is ydydxsinx=cosx(sinx−y2)
ydydx+(cotx)y2=cosx Put y2=z ⇒dzdx=2ydydx ⇒12dzdx+zcotx=cosx ⇒dzdx+2zcotx=2cosx which is a linear differential eqn with z as dependent variable. Here, P=2cotx,Q=2cosx Integrating factor I.F.=e∫Pdx =e∫2cotxdx I.F.=e2logsinx ⇒I.F.=sin2x Solution of given differential eqn is zsin2x=∫2cosxsin2xdx Put sinx=t ⇒cosxdx=dt ⇒zsin2x=2t33+C ⇒y2sin2x=23sin3x+C ⇒y2=23sinx+Csin2x C can be positive or negative. Hence, both options are correct