Separation of variables and IVP's??

Question

Use separation of variables to obtain the solution to the following IVP.

y' = t * y * e^(-t),

y(0) = e^(-1)

y(t) = ?

Answer 1

Divide both sides by y:

y'/y = te^(-t)

Integrate both sides, obtaining an arbitrary constant. the antiderivative of the right side can be found by integration by parts (or from an integral table or a computer algebra system)


ln|y| = C + antiderivative of te^(-t)

After you have done the antidifferentiation, substitute the initital conditions y= e^(-1) and x=0 to evaluate C. Since the initial value of y is positive, the diferential equation shows
y' >0 for t>0, so y will remain positive. Thus you can drop the absolute value sign and then solve for y.

Answer 2

t(dy/dt) = 4y the excellent option dy/4y = a million/t dt combine on the two aspects, u get 0.25ln(4y) = ln(t) + c u merely plug in to get c... 0.25ln(8) = c + 0, c = 0.25ln(8) so u have 0.25ln(4y) = ln(t) + 0.25ln(8) and thats ur answer,,,until you want to resolve for y explicitly..