Semester 3 / Differential Equations / Semester olving-ordinary-differential-equations

Solving Ordinary Differential Equations

1 min read Updated Fri Apr 24 2026 03:19:45 GMT+0000 (Coordinated Universal Time)
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ODEs Can be solved algebraically by using Laplace transform.

Steps

  1. Laplace transform each term using derivative rules.
  2. Substitute initial conditions directly.
  3. Solve algebraic equation in ss-domain.
  4. Apply inverse Laplace transform.

The above steps can be applied for a system of ODEs as well.

Real-World Applications

LCR Circuits

Differential equation:

Ldidt+Ri(t)+1C∫0ti(x)dx=v(t)L \frac{di}{dt} + Ri(t) + \frac{1}{C}\int_0^t i(x)dx = v(t)

Laplace converts it into an algebraic relation involving ( I(s) ) and ( V(s) ).

LTI Systems

Impulse response h(t)h(t), transfer function H(s)=Lh(t)H(s) = L{h(t)}.