Question

Help with Integrating factor in ODE question

2t(dy/dt) + y = t^5

My question is about the Integrating factor = e^(∫dt/(2t)) = e^(ln(t^1/2)) = t^(1/2)

WHY? I thought ∫dt/(2t) =1/2ln|t| why is it t^(1/2)?

Answer 1

\[\huge e^{\int\limits \frac{dt}{2t}}\]

Looking at JUST the integral part, yes it gives you \(\frac{1}{2}ln|t|\),
But rememberrrrr, that was all up in the exponent! :)\[\huge e^{1/2 \ln t}\]Applying a rule of logarithms allows us to bring the 1/2 inside the log,\[\huge e^{\ln (t^{1/2})}\]

Answer 2

From here, remember that the exponential and the logarithm are INVERSE operations of one another.

At this point they essentially cancel out.

Answer 3

\[\huge e^{\ln (t^{1/2})}= t^{1/2}\]

Answer 4

\[\int \frac{dt}{2t}=\frac{1}{2}\int \frac{dt}{t}=\frac{1}{2}ln(t)+c=ln(t^2)+c\]

Answer 5

i meant 1/2

Answer 6

\[ln(t^{1/2})+c\]

Answer 7

then just use zepdrix logarithmic property he showed

Answer 8

wow thank you so much !

Answer 9

also you don't need the c since it'll eventually be added to the other constant to form one ibg constant

Answer 10

cause you have to integrate the right side also and that'll have a constant