# 2[sup]nd[/sup] Order ODE

2nd Order ODE

y''+y=0

I come up with the solutions of y1=c1eix, y2=c2e-ix. Now, using these I try to find the cooresponding real-valued solutions:

eix=cos(x)+isin(x)

e-ix=cos(x)-isin(x)

Both of which have real-valued solutions cos(x). However, when I looked this up online, Wikipedia stated that the answer was y=c1sin(x)+c2cos(x). Am I doing something wrong here? Thanks.

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both $$y_{1} = c_{1} e^{ix}$$ and $$y_{2} = c_{2} e^{-ix}$$ are solutions
Also when you add them up that too is a solution.
So when you add both the Y1 and Y2 solutions you should get the solution on wikipedia

Integral
Staff Emeritus
Gold Member
apmcavoy said:
y''+y=0

I come up with the solutions of y1=c1eix, y2=c2e-ix. Now, using these I try to find the corresponding real-valued solutions:

eix=cos(x)+isin(x)

e-ix=cos(x)-Iain(x)
What you are stating here is Euler's Identity, it is not derived from the solution of the differential equation but is the starting point for going from the imaginary exponential solution to the real valued trig solution.

Both of which have real-valued solutions cos(x). However, when I looked this up online, Wikipedia stated that the answer was y=c1sin(x)+c2cos(x). Am I doing something wrong here? Thanks.
All that separates you from the trig solution is Euler's and some algebra. Realize that the constants in the final solution will not be the same as the constants in your original solution.

stunner5000pt said:
both $$y_{1} = c_{1} e^{ix}$$ and $$y_{2} = c_{2} e^{-ix}$$ are solutions
Also when you add them up that too is a solution.
So when you add both the Y1 and Y2 solutions you should get the solution on wikipedia

Unfortunately I don't. Adding those together gives the following:

$$c_1e^{ix}+c_2e^{-ix}=\left(c_1+c_2\right)\cos{x}+i\left(c_1-c_2\right)\sin{x}=A\cos{x}+Bi\sin{x}$$

...which is not a real solution. Any ideas?

Thanks again.

lurflurf
Homework Helper
if
y=a*exp(i*x)+b*exp(-i*x)
let
c=(a+b)/2
d=(a-b)/(2*i)
then
y=c*cos(x)+d*sin(x)
if you presume x real
y=Re[y]+i*Im[y]
={(Re[a]+Re)*cos(x)-(Im[a]sin(x)-Re)*sin(x)}+i*{(Im[a]+Im)*cos(x)+(Re[a]-Re)*sin(x)}
so for Im[y]=0 (real solution) it is required that
Im[a]+Im=0
Re[a]-Re=0
which is clearly equivalent to
Im[a+b]=0
Re[a-b]=0

Hurkyl
Staff Emeritus
Gold Member
$$c_1e^{ix}+c_2e^{-ix}=\left(c_1+c_2\right)\cos{x}+i\left(c_1-c_2\right)\sin{x}=A\cos{x}+Bi\sin{x}$$
It looks like a real solution to me... (if you choose A and B properly...)

Hurkyl said:
It looks like a real solution to me... (if you choose A and B properly...)

This seems strange to me. Let's say that B=k/i, which would cancel out the i in front of the sine. But now the coefficient isn't a real number. How can a solution be real if it contains complex numbers?

lurflurf
Homework Helper
apmcavoy said:
This seems strange to me. Let's say that B=k/i, which would cancel out the i in front of the sine. But now the coefficient isn't a real number. How can a solution be real if it contains complex numbers?
If the imaginary part is 0
ie the imaginary numbers cancel
is
1+i-i
real?
i/i
e^(pi*i)
i^2
i^i
0*i
cos(i)
real?
2cos(pi/9) is a root of x^3-3x-1
the other roots are 2cos(7pi/9) and 2cos(13pi/9)
thus all are real.
The roots can be expressed in radicals, but only if complex numbers are used.

Hurkyl
Staff Emeritus