# Diff Eq Final

version from 2016-06-16 07:44

## Section

Nullclineswhere dy/dx=0 or -dx/d=0
Separable equations1) Separate equations
2) integrate
3) employ initial condition to find C
General solution of Linear ODEstandard form: y'+P(x)y=g(x)
1) μ(x)=e^(∫p dx)
2) y=1/μ ∫μg + c/μ
Test for exactnessM(x,y)dx + N(x,y)dy=0
Exact if dM/dy = DN/dx
Solve exact equationCombine: f=∫Mdx+k(y)=C &
f=∫Ndy+h(x)=C
Homogeneous?Homogeneous if: M(tx,ty)dx + N(tx,ty)dy=t^p[M(x,y)dx+N(x,y)dy]
To solve homogenous eq.1) let y=ux & dy=udx+xdu. Substitute.
2) separate variables dx&dy
3) integrate
4) sub y/x back in for u, solve for y
BernouliStandard form: y'+Py=gy^n
1) turn into: (y^-n)y'+P(y^1-n)=g
2) substitute v=y^1-n &
v'=(1-n)(y^-n)y'
3) solve like linear ODE
Second Order IVP uniquenessa2y''+a1y'+ay=0 y(0)=p
unique if coefficients are continuous and a2(0)≠0
Linearly Independent or DependentWronskian
w(0): plug in 0 for x
If wronskian=0, functions are L.D
Find second linearly independent solutionStandard form: y''+py'+Qy=0, y1=...
y2=y1∫(e^(-∫pdx))/(y1)^2
General sol: y=c1y1+c2y2
Gen. Sol. of ODE (undetermined coefficients)1) sub. r=y (primes become exponents)
find r1&r2
2) yh=c1cos(xr1)+c2sin(xr2)
3) yp in form of right side
4) derive & plug yp into left side
5) find coefficients
6) y=yh+yp
Gen. Sol. of ODE (Cauchy-Euler)form: x^2y''+axy'+by=0
1) sub. y=x^r & find roots
2)y=x^r[c1+c2ln(x)]
Laplace IVPy''(t)=ay'(t)+by(t)=f(t) y(0)=y0 y'(0)=y1
[(s^2)Y-sy0-y1]+a[sY-y0]+bY=F(s)
Mass dampedmx''+bx'+kx=0
m=mass
b=damping force
k=spring constant