The graph of the solution of the given initial value problem is know to pass thr

Question

The graph of the solution of the given initial value problem is know to pass through the (t,y) points listed. Determine the constants and y0

y'+y=0, y(0)=y0. Solution graph passes through the points (1,4) AND (3,1).

I know the answer but I need help on how to solve it. the answers are =ln(2) and y0=8.

Please show every step and do not combine steps.

Explanation / Answer

y'+ay=0 multiply by e^at (integrating factor) e^at(y'+ay)=0 d/dt( e^at y)=0 => e^at y =C (constant) given y(0)=y0 => 1.y0 = C ... C=y0 It passes through (1,4)=> y(1)=4 e^a1 4= C and e^a3 .1 =C comparing, we get e^(3a-a)=4 => a=0.5*ln4 => a=ln 2 substitute this to get C C= e^3*(ln2) = e^ ln(8) C=y0=8

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