Differential Equations

First order

Separable

Separate and integrate both sides

Linear

Standard form:

\frac{dy}{dx}+p(x)y=q(x)

r(x)=e^{ \int p(x)}

\frac{d}{dx}r(x)y=r(x)q(x)

Second order

Linear Homogeneous

Standard form:

\frac{d^2}{dx^2}y+a\frac{d}{dx}y+by=0

Assume: y=Ce^{mx} ( m may be real or complex)

C(m^2+am+b)Ce^{mx}=0

Since C\ne0 , and e^{mx}>0

m^2+am+b=0

So the solution:

y=Ce^{m_1x}+De^{m_2x}

  • Two real m
  • Two complex m
  • One real m

Two complex m

As m_{1,2} is complex numbers, they can be rewritten as

m_{1,2}=a\pm ik

y=e^ax(Ce^{ikx}+De^{-ikx})

Using Euler’s theorem,

y=e^{ax}(Acos kx+Bsin kx)

One real m

y=(A+Bx)e^{mx}

Coupled DE

  • \frac{dx}{dt}=f(x, y, t)
  • \frac{dy}{dt}=g(x, y, t)

Coupled Linear DE with constant coefficients

\frac{dx}{dt}=ax+by (1)
\frac{dy}{dt}=cx+dy (2)

Differentiate (1) wrt t (make it second order derivative of x wrt t )

Substitute for \frac{dy}{dt} and y (getting rid of y )

Solve DE wrt x