• Author / Uploaded
• ɹɐʞ Thye

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Method For solving linear equations (a) Write the equation in the standard form

(b) Calculate the integrating factor u(x) by the formula ∫

(c) Multiply the equation in standard form by u(x) and, recalling that the left-hand side is just

, obtain

(d) Integrate the last equation and solve for y by dividing by u(x) to obtain ∫ Problem 1 A rock contains two radioactive isotopes, RA1 and RA2, that belong to the same radioactive series; that is RA1, decays into RA2, which then decays into stable atoms. Assume that the rate at which RA1 decays into RA2 is kg/sec. Because the rate of decay of RA2 is proportional to the mass y(t) of RA2 present, the rate of change in RA2 is

where k>0 is the decay constant. If k = 2/secand initially y(0) = 40kg, find the mass y(t) of RA2 for t≥0.

Solution Since equation (1) is linear, so we begin by writing it in standard form

Given k = 2 and initial y(0) = 40, Substitute k=2 into equation (2) and displayed the initial condition.

From equation (3), P(t) = 2, ∫ ∫

Multiplying equation (3) by

,

Integrating both sides, ∫

Then solve the equation for y,

∫

Given y(0) = 40 and substituting t =0,

So C =

. Thus, the mass y(t) of RA2 at time t is given by