1.

Find the differential equation of the family of curves `y=A e^(2x)+B e^(-2x)`, where A and B are arbitrary constants.

Answer» `y=A tan^(-1)(B tan""(x)/(2)),`
`"where "A=(2)/(sqrt(a^(2)-b^(2))),B=sqrt((a-b)/(a+b))`
`Ab=(2)/(sqrt((a-b)(a+b)))sqrt((a-b)/(a+b))=(2)/(a+b)`
`(dy)/(dx)=(ABsec^(2)""(x)/(2)xx(1)/(2))/(1+B^(2)tan^(2)""(x)/(2))`
`=(1)/(a+b).(a+b)/((a+b)cos^(2)""(x)/(2)+(a-b)sin^(2)""(x)/(2))`
`=(1)/(a+b cos x)" (1)"`
`therefore" " (d^(2)y)/(dx^(2))=(b sin x)/((a+b cos x)^(2))`


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