使用者:Tigerzeng/常見常微分方程的通解

The general solution to a differential equation is the most general form that the solution can take and does not take any initial conditions into account. Solving for the general solution of a differential equation (that is solvable) is usually the major concern of a student. Here are the general solutions of some of the most common types of ordinary differential equations (ODE) anybody would encounter in a differential equation class.
Note: Integration constant can be omitted in all of the Integrating Factors (IF) below while the integration part in the general solution requires the "+C".

First-order ODE:
1. Linear ODE

  Form: y'+ B(x)y = C(x)
  IF: μ = e∫B(x)dx
  General Solution: y = μ-1 • ∫C(x)μdx
  Note: Coefficient of y' must be 1. 

2. Exact ODE

  Form: M(x,y)dx + N(x,y)dy = 0    OR    M(x,y) + N(x,y)dy/dx = 0   ,where the condition My = Nx has to be fulfiled first.
  IF: h(y) = ∫(N - ∂y∫Mdx) dy
  General Solution: ∫Mdx + h(y) = c  ,where c is a constant.
  Note: Write ∫Mdx first.

3. Clairaut's Equation

  Form: y = xy' + B(y')
  IF: None
  General Solution: y = cx + B(c)   ,where c is a constant.
  Singular Solution: x = -B'(t) , y = B(t) + tx
  Note: General solution is a family of straight lines.
        Singular solution exists when B(2)(t)≠0.
        Singular solution is the envelope of general solution and is in form of parametric equation.

4. Lagrange's Equation

  Form: y = xA(y') + B(y')
  Substitute: y' = t
  IF: μ = e∫A'(t)/(A(t)-t)dt
  General Solution: x = μ-1 • ∫μB'(t)/(t-A(t))dt
                    y = xA(t) + B(t)
  Singular Solution: y = A(c)x + B(c)   ,where c is the root of A(t)=t.
  Note: When A(t) = t, it becomes Clairaut's Equation, so Clairaut's Equation is a specific form of the Lagrange's Equation.

5. Riccati Equation

  Form: dy/dx = A(x)y2 + B(x)y + C(x)
  IF: μ = e∫2A(x)y1 + B(x)dx
  General Solution: y1 - μ • (∫A(x)μdx)-1   ,where y1 is a particular solution (Either it is given in the question or it is by guessing.)
  Note: There are two types of common substitution. First is substitute u = y1 + v-1. Second substitution is y = -k'/Ak. But the second substitution often leads to a second-order, non-homogeneous, non-constant coefficient ODE, which is mostly very hard or even impossible to solve.

6. Almost Exact ODE

  Form: same form with exact ODE but do not fulfil the condition My = Nx
  IF: μ(x) = e∫(My-Nx)/N dx  [1]  OR    μ(y) = e∫(Nx-My)/M dy  [2]  OR    μ(x,y) = e∫(Nx-My)/(xM-yN) d(xy)  [3]
  General Solution: same general solution with exact ODE. After finding the IF, multiply the whole equation with IF and it should turn back to a exact ODE.
  Note: For μ(x) and μ(y), pick the easier one between M and N as the denominator and remember two things. Firstly, it is always Nx and My. Secondly, the end of the numerator and the differential is always the same with the denominator.
        For μ(x,y), the integration part should be with respect to xy.

7. Bernoulli Equation

  Form: y' + B(x)y = C(x)yn
  IF: μ = e(1-n)∫B(x)dx
  General Solution: y = (μ-1•(1-n)• ∫C(x)μdx)1/(1-n)
  Note: When n=0 or n=1, it becomes a linear ODE

Second-order system of ODEs:
Complex eigenvalues

  For complex eigenvalues λ = r±θi and eigenvector v=[a1+b1i   a2+b2i]T
  Define function Kn = det([ [an   bn]T   [-c2   c1]T ])  and  Jn = det([ [an   bn]T   [c1   c2]T ])
  

  General solution: x(t) = ert • [ [K1   K2]T   [J1   J2]T ]  •  [cosθt   sinθt]T   &  [K1   K2]T = x(0) = x0   (usually it would be x(0) = x0 as the inital condition) and c1, c2 are the constants for the initial value problem.