Linear First Order ODE

A linear first order ODE is a differential equation that is express as a linear combination of a function and it’s derivatives. It can be reduced into a Separable ODEs using an integrating factor.

Definition §

A linear first order ODE is of the form:

Linear First Order ODE

$$\frac{\text{d}y}{\text{d}x}f(x)+g(x)y+h(x)=0$$

Note

• $f(x),g(x)$ = Functions depending solely on $x$
• $h(x)$ = The forcing term
• If $h(x)=0$, the ODE is homogenous

Integrating Factor §

To solve these ODEs, the left hand side needs to be represented as a product, like $A(x)B(y)=\frac{\text{d}y}{\text{d}x}$, then it is separable. In order to do that, we use the product rule to find an integrating factor $\mathcal{I}(x)$:

$$\mathcal{I}(x)=e^{\int f(x)dx}$$

Why?

We need $\mathcal{I}(x)$ such that: $\frac{d}{dx}(\mathcal{I}\times y)=\frac{dy}{dx}\mathcal{I}(x)+f(x)\mathcal{I}(x)y$

Using Product rule $\frac{d\mathcal{I}}{dx}y+\frac{dy}{dx}\mathcal{I}=\frac{dy}{dx}\mathcal{I}+f(x)\mathcal{I}(x)y$ $\to \frac{d\mathcal{I}}{dx}\cancel{y}+\cancel{\frac{dy}{dx}\mathcal{I}}=\cancel{\frac{dy}{dx}\mathcal{I}}+f(x)\mathcal{I}(x)\cancel{y}$ $\to \frac{d\mathcal{I}}{dx}=f(x)\mathcal{I}(x)$ Separable ODEs $\to \int \frac{1}{\mathcal{I}}d\mathcal{I}=\int f(x)dx$ $\to \mathcal{I}=e^{\int f(x)dx}$

Since we only need one factor, we can let the constant of integration, c, equal to zero.

Then, simply multiply the entire ODE by $I(x)$:

$\frac{dy}{dx}I(x)+f(x)I(x)y=-g(x)I(x)$ $\to \frac{d}{dx}(y\times I(x))=-g(x)I(x)$ $\to y=\frac{1}{I(x)}\int -g(x)I(x)dx$

Homogenous §

When $h(x)=0$, the ODE is considered homogenous, with a solution given by:

$$y(x)=\frac{1}{\mathcal{I}(x)}\int -g(x)\mathcal{I}(x)\text{d}x$$

where $\mathcal{I}(x)$ is the integrating factor:

Inhomogeneous §

When $h(x)\ne 0$ , the ODE is considered inhomogeneous, with a solution:

$$y(x)=y_h(x)+y_p(x)$$

• $y_h(x)$ is the homogenous solution to the ODE, also known as the transient solution
• $y_p(x)$ is the particular solution to the inhomogeneous version.