## haganmc 3 years ago find particular solution to 2x' + x=3t^2

1. amistre64

what methods do you know of?

2. haganmc

undetermined coefficients

3. amistre64

got a lousy connection i believe

4. Algebraic!

happening to me too, OS is bugged today.

5. JakeV8

site is slow for me too :(

6. amistre64

well, lets start by dividing off that 2 x' + x/2 = 3t^2/2 we want to determine an e^rt that will allow us to turn this into a product rule; e^t/2 should work e^(t/2) x' +e^(t/2) x/2 = e^(t/2)3t^2/2 when we integrate both sides we get $e^{t/2}x= \int e^{t/2}\frac{3t^2}{2}dt$

7. Algebraic!

trial soln. is at^2 + bt +c 2x' = 4at + 2b x= at^2 +bt +c a=3 4a+b =0 2b+c =0 b=-12 c=24

8. amistre64

$\int e^{t/2}\frac{3t^2}{2}dt$ e^(t/2) +t^2 2e^(t/2) -2t 4e^(t/2) +2 8e^(t/2) $2*\frac32e^{t/2}(t^2-4t+8)$

9. amistre64

forgot the +C at the end:) this gives us $e^{t/2}x=3e^{t/2}(t^2-4t+8)+C$ divide off the e^{t/2} $x=3(t^2-4t+8)+Ce^{-t/2}$

10. amistre64

my idea was undetermined coeffs tho was it

11. amistre64

*wasnt .... ugh, when things slow down, they sloowww down

12. Algebraic!

I did it the 'integrating factor' way too, just because I was curious... fair bit of work doing that integral..