The differential equation whose general solution is given by, y = (c1cos(x+c2))−(c3e−x+c4)+(c5sinx), where c1,c2,c3,c4,c5 are arbitrary constants, is
A
d4ydx4−d2ydx2+y=0
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B
d3ydx3+d2ydx2+dydx+y=0
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C
d5ydx5+y=0
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D
d3ydx3−d2ydx2+dydx−y=0
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Solution
The correct option is Bd3ydx3+d2ydx2+dydx+y=0 y=c1cos(x+c2)−(c3e−x+c4)+(c5sinx)⇒y=c1(cosxcosc2−sinxsinc2)−(c3ec4e−x)+(c5sinx)⇒y=(c1cosc2)cosx−(c1sinc2−c5)sinx−(c3ec4)e−x⇒y=lcosx+msinx−ne−x............(1)
where l, m, n are arbitrary constant ⇒dydx=−lsinx+mcosx+ne−x..............(2)⇒d2ydx2=−lcosx−msinx−ne−x.............(3)⇒d3ydx3=lsinx−mcosx+ne−x..............(4)Fromequations(1)+(3),d2ydx2+y=−2ne−x.............(5)Fromequations(2)+(4),d3ydx3+dydx=2ne−x.............(6)Fromequations(5)+(6),wegetd3ydx3+d2ydx2+dydx+y=0