Linear Circuits 1 - 24 - Differential Equations

Explore how differential equations describe how voltages and currents in linear circuits change over time, extending Ohm's law to time-varying resistance.

Explore how resistance can vary with time through ohm's law and inductance, showing how voltage and current evolve in r-l-c circuits with capacitors described by second-order differential equations.

Explore the parts of a differential equation, linking output y(t) with past, present, and future via derivative and integral terms, and show how input x(t) drives the system.

This lecture introduces integral differential equations and demonstrates turning them into purely differential equations by differentiation, using the capital D operator to write derivatives and highlighting easier solving for engineers.

Learn to solve differential equations via zero input response, using D operator notation with Q(D) and P(D), illustrated by a circuit with a zero voltage source and an inductor.

Show how a second‑order differential equation yields a characteristic polynomial by substituting y = C e^{\lambda t}; solve λ^2+3λ+2=0 to get λ1, λ2 and the system’s response.

Solve the characteristic equation to obtain eigenvalues that define the exponential characteristic modes e^(λ t). Express the solution as a combination of these modes, describing decay, settling, or decaying oscillations.

Solve second-order differential equations in zero-input response using exponential characteristic modes; determine eigenvalues from the characteristic equation, apply initial conditions y(0) and y'(0) to find constants.

Solve for C1 and C2 using y(0)=0 and y'(0)=-1 to obtain the specific solution y(t) = -1/2 e^{-t} + 1/2 e^{-3t} for the given differential equation.

Explain how the zero-input differential equation y''+4y'+3y=0 relates to exponential modes e^{-t} and e^{-3t}, and how initial conditions set C1 and C2 so the output decays to zero.

See how the same differential equation produces different outputs with varying initial conditions, solving for C1 and C2 to give trajectories like y(t)=2e^{-t}-e^{-3t} that decay to zero.

Explore special cases of solving linear differential equations via eigenvalues: distinct real, repeated real, and complex conjugate roots, with their corresponding general solutions and initial-condition requirements.

Read the polynomial Q(D) from the differential equation, form the characteristic equation, and solve for eigenvalues; distinct, repeated, or complex conjugate values govern zero input response with decay and oscillation.

Solve a linear circuit with a zero voltage source using differential equations; derive the current y(t) from an initial 1 A inductor, giving y(t)=C e^{-2t/3} fixed by the initial condition.