Consider the differential equation for the function y given by, y\' = 4y + 4t/t,
ID: 2879577 • Letter: C
Question
Consider the differential equation for the function y given by, y' = 4y + 4t/t, y(1) = 4. a. Transform the equation for _y above into a separable equation for a function nu. If we write the equation as v'(t) = f(t, nu), then find f(f, v). Important: Since there are many functions v with the property above, choose it as in Section 1.3 of the Lecture Notes. f(t, v) = (3v+4)(1/t) b. Find the general solution nu_g to the equation found in part (a). Denote by c the integration constant. v_g(t) = c. Use the function v found in part (b) to find the unique function y solution to the initial value problem above. y(t) =Explanation / Answer
b> f(t,v) = v'(t) = dv/dt = (3v+4)(1/t)
=> dv/(3v+4) = dt/t
integrating both sides
=> ln(3v+4)/3 = ln(t) + K
3v+4 = e^3ln(t)*e^(3K)
3v+4 = t^(3)e^(3K)
v = t^3*e^(3K)/3 - 4/3
vg(t) = ct^3 - 4/3 ,---------->this is the required general solution