2nd in class week 7 Convert the 2nd order ODE into two 1st o

2nd in class week 7 Convert the 2nd order ODE into two 1st order ODEs 2y\" + 3y\' = sin2t Are the following functions linearly independent on the given interval? Give your reason. sec2 x, cos2 x for 0LE xl

Solution

2y\"+y\' = sin²t

Substitute dy/dt = x

Then d^y/dt^2 = dx/dt

Hence the equation becomes 2dx/dt + x = sin²t

This is the first order equation and can be solved.

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in 0<x<pi/2

cos square cannot be written as a linear combination of sec^2 x

Hence they are linearly independent

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2y\" +3y\'-3y = sin²t

Put y\' =x

y\" = x\'

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5x^4, e^4lnx

are equivalent to 5x^4 and x^4

Hence they are linearly dependent.