Consider the IVP for the function y given by -2 cos(3t) y\' - 2t^4 e^yy\' + 6y s

Question

Consider the IVP for the function y given by -2 cos(3t) y' - 2t^4 e^yy' + 6y sin (3t) - 8t^3 e^y = 0, y(0) = 1 Just by reordering terms on the left hand side above, write the equation as N y' + M = 0 for appropriate functions N, M. If the differential equation is exact, find a potential function psi, such that any solution y satisfies psi(t, y) = C. If the equation is not exact, type Not Exact. If the equation is exact, find the unique potential function psi, such that any solution y of the IVP satisfies psi(t, y) = 0. If the equation is not exact, type Not Exact.

Explanation / Answer

a) consider -2cos(3t)y' - 2t^4e^yy' + 6ysin(3t)-8t^3e^y = 0, y(0)=1.

N(t,y) = -2cos(3t)y'-2t^4e^yy'

M(t,y) = 6ysin(3t)-8t^3e^y

given, Ny' + M = 0

N(t,0) = -2cos(3t)(0) - 2t^4 * e^0 *0

N(t,y) = 0

M(t,0) = 6*0*sin(3t) - 8t^3 * e^0
= -8t^3*1
M(t,y) = -8t^3

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