Differentiating First-Order Linear ODEs with an Integrating Factor

[hansbahia]

Solve the differential equations specified using an integrating factor.
a) Find the general solution of the differential equation
dy/dt = -2ty + 4e^(-t^2)
b) Solve the initial-value problem
dy/dt = 1/(t+1)y + 4t^2 + 4t, y(1) = 10

 

[HallsofIvy]

You were asked to read rules for this forum when you registered and it appears that you did not do that. First, you titled this "Differential Homework" (which was good of you) but did not post it in the homework section. Second, you have shown no attempt to solve the problem yourself- we will be happy to help you do your homework but we will not do it for you. And seeing what you have done and where you get stuck or go wrong will help us give hints and suggestions.

Now, the problem says "using the integrating factor". Do you know what an "integrating factor" is? There is standard method of finding an integrating factor for a linear equation and both of these equations are linear. Do you know that method?

I will move this to the homework section for you.

 

[hansbahia]

Sorry I forgot,
let me ask it again...
About using integrating factor, how can I find the general solution of the differential equation, let’s use a different equation dy/dt = -2ty + 4e^(-t^2)
Also how to solve the initial-value problem dy/dt = 1/(t+1)y + 4t^2 + 4t, y(1) = 10

I know that is an integrating factor is a function that is selected to make easy the solving of a given equation involving differentials. Also, I know, for example if we have y’+P(x)y=Q(x), after doing all the math the integrating factor would be equal e^(Integral of P(x)dx). But how do I do, when the equations are complicated like above

 

[hansbahia]

Already,

so dy/dt = 1/(t+1)y + 4t^2 + 4t, y(1) = 10

by using the integrating factor(IF)

IF= e^(integral of P(x))
P(x)= 1/(t+1), so IF=e^(1/(t+1))=e^(ln(t+1))=t+1

now that I have the integrating factor I multiply everything by t+1

dy/dt = 1/(t+1)y + 4t^2 + 4t
(t+1) dy/dt = (t+1)(1/(t+1))y + (t+1)4t^2 + (t+1)4t
multiplying...
(t+1)dy/dt = y + 4t^3+4t^2 + 4t^2+4t
adding the ts...
(t+1) dy/dt = y + 4t^3 + 8t^2+4t, 4t^3 + 8t^2+4t= (t+1)(4t^2+4t)
dividing both sides by t+1

(t+1)/(t+1) dy/dt = (y + (t+1)(4t^2+4t))(t+1)

simplifying...

dy/dt= y/(t+1)+4t^2+4t

Now how can I differentiate this first-order linear ordinary equation?