Linear Differential Equations | Ch.9 Differential Equations

📐 Standard Form of a Linear DE

Definition

A first-order linear differential equation is of the form:

Standard Form $$\frac{dy}{dx} + P(x)\,y = Q(x)$$

where $P$ and $Q$ are continuous functions of $x$ (not $y$).

The key feature: $y$ appears only to the first power, and $y$ and $y'$ are not multiplied together.

Note: The equation can also be written in terms of $x$ as a function of $y$: $\dfrac{dx}{dy} + P_1(y)\,x = Q_1(y)$, where $P_1$ and $Q_1$ are functions of $y$ only.

🔑 The Integrating Factor

Integrating Factor (IF)

Multiplying both sides of the DE by a special function called the Integrating Factor makes the left side a perfect derivative:

Integrating Factor $$\text{IF} = e^{\int P(x)\,dx}$$

After multiplying by IF:

\frac{d}{dx}\bigl[y \cdot \text{IF}\bigr] = Q(x) \cdot \text{IF}

🪜 Method of Solution

Write the DE in standard form: $\dfrac{dy}{dx} + Py = Q$
Identify $P(x)$ and $Q(x)$.
Find the Integrating Factor: $\text{IF} = e^{\int P\,dx}$
Multiply both sides by IF.
Integrate both sides: $y \cdot \text{IF} = \displaystyle\int Q \cdot \text{IF}\,dx + C$
Simplify to get the general solution.

Complete Solution Formula $$y \cdot e^{\int P\,dx} = \int Q\cdot e^{\int P\,dx}\,dx + C$$

💡 Worked Examples

Example 1

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

$P = 1,\quad Q = e^x$

IF $= e^{\int 1\,dx} = e^x$

Multiply: $e^x\dfrac{dy}{dx} + e^x y = e^{2x}$, i.e., $\dfrac{d}{dx}(ye^x) = e^{2x}$

Integrate: $ye^x = \dfrac{e^{2x}}{2} + C$

\boxed{y = \frac{e^x}{2} + Ce^{-x}}

Example 2

Solve: $\dfrac{dy}{dx} - \dfrac{y}{x} = x^2$

$P = -\dfrac{1}{x},\quad Q = x^2$

IF $= e^{\int -1/x\,dx} = e^{-\ln x} = \dfrac{1}{x}$

Multiply: $\dfrac{d}{dx}\!\left(\dfrac{y}{x}\right) = x$

Integrate: $\dfrac{y}{x} = \dfrac{x^2}{2} + C$

\boxed{y = \frac{x^3}{2} + Cx}

Example 3 — DE in $x$ and $y$

Solve: $\dfrac{dx}{dy} + x = y^2$

$P_1 = 1,\quad Q_1 = y^2$ (functions of $y$)

IF $= e^{\int 1\,dy} = e^y$

Multiply and integrate: $xe^y = \displaystyle\int y^2 e^y\,dy$

Using integration by parts: $\displaystyle\int y^2 e^y\,dy = e^y(y^2-2y+2)$

\boxed{x = y^2 - 2y + 2 + Ce^{-y}}

📊 Quick Reference

Form	IF	Solution
$\frac{dy}{dx} + Py = Q$	$e^{\int P\,dx}$	$y \cdot \text{IF} = \int Q \cdot \text{IF}\,dx + C$
$\frac{dx}{dy} + P_1 x = Q_1$	$e^{\int P_1\,dy}$	$x \cdot \text{IF} = \int Q_1 \cdot \text{IF}\,dy + C$

🔑 Key Takeaways

Standard form: $\frac{dy}{dx} + P(x)y = Q(x)$
Integrating Factor: $e^{\int P\,dx}$ — never contains arbitrary constant.
Multiply both sides by IF, then integrate directly.
The DE can also be in the form $\frac{dx}{dy} + P_1 y = Q_1$ (swap roles of $x$ and $y$).
This is the most powerful and frequently used method in Chapter 9.