- #1
freezer
- 76
- 0
Homework Statement
Solve the initial value problem:
[tex]
x^{2}{y}'' + x{y}' + y = 0, x>0, y(1)=1, {y}'=2
[/tex]
Homework Equations
y=x^m
The Attempt at a Solution
[tex]
x^{2}(m(m-1)x^{m-2})+xmx^{m-1} + x^{m}
[/tex]
[tex]
x^{2}(m(m-1)x^{m-2})+xmx^{m-1} + x^{m}
[/tex]
[tex]
x^{m}(m(m-1) + m + 1)
[/tex]
[tex]
m = \pm i
[/tex]
This is the way i was doing it:
[tex]
C_1 e^{it} + C_2 e^{-it}
[/tex]
[tex]
C_1(cos(t) + i sin(t)) + C_2(cos(t) - i sin(t))
[/tex]
The solution shows:
[tex]
C_1 x^{i} + C_2 x^{-i}
[/tex]
[tex]
C_1(cos(ln(x)) + i sin(ln(x))) + C_2(cos(ln(x)) - i sin(ln(x)))
[/tex]
With the initial conditions indicate that the solution is correct. Yet the textbook shows the form to be:
[tex]
Y(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x}
[/tex]
not
[tex]
Y(x) = C_1 x^{r_1} + C_2 x^{r_2}
[/tex]
And if the second form is correct, I have not found an Euler identity that supports the step.