e^(t)*y dy/dt = e^(-y) + e^(-2t - y)

ye^t dy/dt = e^(-y) + e^(-(2t +y))

Put them all in positive exponents,

ye^t dy/dt = 1/(e^y) + 1/(e^(2t +y))

ye^t dy/dt = 1/(e^y) + 1/[(e^(2t))(e^y)]

Make the two fractions in the RHS into one fraction only,

common denominator is e^(2t) *e^y,

ye^t dy/dt = [e^(2t) +1] / [(e^(2t))(e^y)]

Clear the fraction, multiply both sides by [(e^(2t))(e^y)],

[(e^(2t))(e^y)][ye^t] dy/dt = e^(2t) +1

(ye^y)(e^(3t)) dy/dt = e^(2t) +1

(ye^y) dy/dt = [e^(2t) +1] /[e^(3t)]

(ye^y) dy = ([e^(2t) +1] / [e^(3t)])dt

(ye^y)dy = {(e^(2t)/(e^(3t)) +1/(e^(3t))}dt

(ye^y)dy = {1/(e^t) +1/(e^(3t))}dt

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