Basic Differential Equations and Solutions
Separation of variables
f1(x)g1(y)dx + f2(x)g2(y)dy = 0
Linear first order eqaution
dx/dy + P(x)y = Q(x)
dy/dx + P(x)y = Q(x)yn
where v = y1-n.
If n = 1, the solution is
M(x,y)dx + N(x,y)dy = 0, where ∂M/∂y = ∂N/∂x.
where ∂x indicates that the integration is to be performed with respect to x keeping y constant.
dy/dx = F(y/x).
where v = y/x.If F(v) = v, the solution is y = cx.
yF(xy)dx + xG(xy)dy = 0
where v = xy. If G(v) = F(v), the solution is xy = c.
Linear, homogeneous second order eqaution
d2y/dx2 + a(dy/dx) + by = 0 , a,b are real constant.
Let m1, m2 are roots of m2 + am + b = 0.Then there are three cases
Case 1. m1,m2 real and distinct:
Case 2. m1,m2 real and equal:
Case 3. m1 = p + qi,m2 = p - qi:
Linear, nonhomogeneous second order equation
d2y/dx2 + a(dy/dx) + by = R(x), a, b are real constants.
There are three cases corresponding to the above ones.
Euler or Cauchy equation
x2d2y/dx2 + a(dy/dx) + by = S(x) .
Putting x = et, the equation becomes
d2y/dt2 + (a - 1)(dy/dt) + by = S(et)
and can then be solved as the above two entries.
x2d2y/dx2 + x(dy/dx) + (λ2x2 - n2)y = 0.
y = c1Jn(λx) + c2Yn(x).
Transformed Bessel's equation
x2d2y/dx2 + (2p + 1)x(dy/dx) + (α2x2r + β2)y = 0.
where q = √ .
(1 - x2)d2y/dx2 - 2xdy/dx + n(n + 1)y = 0.
y = c1Pn(x) + c2Qn(y).