Hw1
Need maple
Number 3
D2y/dx2=y”
Y”+k^2y=0, Find y(x)
Y=sin(kx), cos(kx) by inspection
Y”-k^2y=0, find y(x)
X^2*y” + x*y’ + k^2*x^2*y = 0
X^2*y” + x*y’ – k^2*x^2*y = 0
d/dx*[(1-x^2)*y’] + n(n+1)y = 0, find y(x)
-----------------------------------------------------------------------------------------------------------X^2 +2x=0, find x, either solution or no solution
Main difference between algebraic and differential equation
- Differential equations include derivatives of the original, algebraic does not
D2y/dx2 = 0  y=C1x+C2
How to solve ODE’s?
- Inspection
- Analytical methods
o Series method
 Y=sum(Ak*x^k), A uknown
 Y=sum[Ak*x^(k+s)], Frobenius, A and s unknowns
- Numerical methods
o Use discretization to convert diffEQ form differential to algebraic
o Approximation
Try plotting sinx for the homework
Finding general solutions to 2nd order ODE’s,
Sin(kx), cos(kx), e^kx, e^-kx, sinh(kx), cosh(kx), Jo(kx), Yo(kx), Io(kx), Ko(kx), Pn(kx),
Qn(kx)
[Characteristic functions, eigen functions]
Special functions as Power Series
Power Series:
Ao+A1x+a2x^2+…
- infinite series, must be truncated to be practical
- most series in practical applications only need to be taken out to about 5 or so
terms
y”-y=0, assume power series, general versions of power series
sum[(k-1)*k*Ak*x^(k-2)]-sum[Ak*x^k]=0
Coefficients of X need to equal 0, all A’s can be written in terms of A0 and A1