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⬅ Back to Differential Equations
📘 Key Concepts: Linear First-Order Equations (Integrating Factor)

Standard form: $\dfrac{dy}{dx} + P(x)\,y = Q(x)$

The integrating factor is $\mu = e^{\int P(x)\,dx}$. Multiply through by $\mu$:

$$\frac{d}{dx}[\mu\, y] = \mu\, Q(x) \quad\Longrightarrow\quad \mu\, y = \int \mu\, Q(x)\,dx + C$$

Steps:

  1. Write in standard form: $y' + P(x)y = Q(x)$.
  2. Compute $\mu = e^{\int P\,dx}$ (no constant needed here).
  3. Multiply both sides by $\mu$; left side becomes $(\mu y)'$.
  4. Integrate both sides, divide by $\mu$, apply initial condition.

Application: RL circuits — $L\dfrac{di}{dt} + Ri = E(t)$ — are linear first-order DEs.

Problem 1: Basic Linear First-Order

Solve $\dfrac{dy}{dx} + y = e^x$.

DE – Linear – Problem 1 – Setup DE – Linear – Problem 1 – Diagram DE – Linear – Problem 1 – Diagram
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Problem 2: Linear with Variable Coefficients and IVP

Solve $\dfrac{dy}{dx} + \dfrac{2y}{x} = x^2$, given $y(1) = 2$.

DE – Linear – Problem 2 – Setup DE – Linear – Problem 2 – Diagram DE – Linear – Problem 2 – Diagram
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Problem 3: Linear with Trigonometric Coefficient

Solve $\dfrac{dy}{dx} - y\tan x = \sec x$.

DE – Linear – Problem 3 – Setup DE – Linear – Problem 3 – Diagram DE – Linear – Problem 3 – Diagram
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Problem 4: RL Circuit Application

An RL circuit has inductance $L = 0.1\,\text{H}$, resistance $R = 10\,\Omega$, and a constant EMF of $E = 50\,\text{V}$. The governing equation is $L\dfrac{di}{dt} + Ri = E$, with initial current $i(0) = 0$.
a. Find $i(t)$.
b. Find the steady-state current (as $t \to \infty$).
c. Find the time at which the current reaches 90% of its steady-state value.

DE – Linear – Problem 4 – Setup DE – Linear – Problem 4 – Diagram DE – Linear – Problem 4 – Diagram
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