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Question

Find the differential equation of the family of curves y=Ae2x+Be2x, for different values of A and B.

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Solution

Given,

y=Ae2x+Be2x

differentiating on both sides, we get,

y=3Ae3x2Be2x

again differentiating on both sides, we get,

y′′=9Ae3x+4Be2x

Now,

y′′y=9Ae3x+4Be2x(3Ae3x2Be2x)

=9Ae3x+4Be2x3Ae3x+2Be2x

=6Ae3x+6Be2x

=6(Ae2x+Be2x)

=6y

y′′y=6y

Is the required differential equation of family of curves.

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