## MathSofiya 3 years ago solve the differential equation using the variation of parameters method $y''+y=sec^2x,0\le x\le \frac{\pi}2$ $r^2+1=0$

1. Algebraic!

did you get that problem from the other day sorted out?

2. Algebraic!

y" +y =t*e^t iirc?

3. MathSofiya

at first I had r^2+r=0 and that gave me the integral of $\int\frac{e^{-x}sec^2x}{e^{-x}}dx+e^{-x}\int \frac{sec^2x}{e^{-x}}dx$

4. MathSofiya

but that is apparently wrong

5. MathSofiya

iirc? what does that mean?

6. Algebraic!

if I recall correctly

7. MathSofiya

$r_1=r_2=\pm1i$

8. MathSofiya

$y_c=c_1cosx+c_2sinx$

9. MathSofiya

$W(y_1,y_2)=cos^2x+sin^2x$

10. Algebraic!

1

11. MathSofiya

nice! $y_p=-cosx\int sinxsec^2xdx+sinx \int cosxsec^2xdx$

12. MathSofiya

so far so good?

13. Algebraic!

yes

14. MathSofiya

$y=y_c+y_p=c_1cosx+c_2sinx-cosxsecx+\int cosxsec^2xdx$

15. Algebraic!

missing a 'sin' on that last term...

16. psi9epsilon

@mathsofiya , check integration seems incomplete

17. Algebraic!

lol

18. psi9epsilon

@Algebraic! , didnt really see the reason for "lol", there are couple of things missing here

19. Algebraic!

"integration seems incomplete"

20. Algebraic!

what tipped you off?

21. Algebraic!

here I thought you were making a funny and you were somehow serious...

22. psi9epsilon

a simple look suggests that last term carries an Integration sign whereas the second last term does not, it cant be both , either we have an integral sign or we dont. Anyways its better to teach something correctly : )

23. MathSofiya

where's a sine missing?

24. Algebraic!

sinx∫cosxsec^2x dx -> ∫cosxsec^2x dx

25. MathSofiya

very true $y=y_c+y_p=c_1cosx+c_2sinx-cosxsecx+sinx\int cosxsec^2xdx$

26. MathSofiya

is this really the solution to the integral? http://www.wolframalpha.com/input/?i=integral+of+cosxsec^2x

27. Algebraic!

most people use ln(tan(x) +sec(x)) for brevity

28. Algebraic!

programs always give the other form of the answer; not sure why, probably to confound students.