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Question

Form the differential equation representing the family of curves y=e2x(a+bx), where a and b are arbitrary constants.

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Solution

y=e2x(a+bx)
It consists of 2 arbitrary constants.
This means we can differentiate it twice to get the differential equation.
Multiply by e2x on both sides, we get
e2xy=a+bx
Differentiating wrt x, we get
e2xdydx+(2)e2xy=be2x(dydx2y)=b
Again, differentiating wrt x, we get
e2x(d2ydx22 dydx)+(2)e2x(dydx2y)=0e2x(d2ydx24dydx+4y)=0d2ydx24dydx+4y=0 [e2x0]
This is the required differential equation.

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