02. First Order Differential Equations

Ordinary Differential Equation

An equation relating a variable $x$ to a function $y$ (with the finitely many derivatives of $y$ with respect to $x$)

Such that it can be written as

$$f\left(x,y,\frac{dy}{dx},\frac{d^{2}y}{d{x}^2},\cdots ,\frac{d^{k}y}{d{x}^k}\right)=0$$

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Order of an ODE

If the order of an ODE is $k$ then

The ODE involves the derivatives of order $k$ or less

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2.1 Direct Integration

First Order Differential Equations

Order Differential Equations in the form of

$$\frac{dy}{dx}=f(x,y)$$

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Direct Integration (for First ODEs)

If the ODE takes the form of

$$\frac{dy}{dx}=f(x)$$

Then we can integrate both sides with respect to $x$ so

$$y=\int f(x)dx+c$$

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2.2 Separation of Variables

Separation of Variables

If the ODE takes the form of

$$\frac{dy}{dx}=a(x)b(y)$$

where $a$ is a function of $x$ and $b$ is a function of $y$ then

$$\frac{1}{b(y)}\frac{dy}{dx}=a(x)$$

Hence by integrating with respect to $x$ then

$$\int \frac{1}{b(y)}dy=\int a(x)dx$$

(Assume that $b(y)\ne 0$ otherwise $y=c\in \mathbb{R}$ is a solution)

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2.3 Reduction to separable form by substitution

Reduction to separable form by substitution

A reduction to separable form is a substitution that transforms a differential equation

$$\frac{dy}{dx}=F(x,y)$$

into an equation where variables can be separated, i.e.

$$g(y)\frac{dy}{dx}=f(x)$$

The goal is to find a substitution ( $u=\varphi (x,y)$ ) that simplifies the relation between ($x$) and ($y$) so that the resulting equation becomes separable.

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Implicit Solutions

Solutions (typically to an ODE) where we have not found a $y$ in terms of $x$

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2.4 First Order Linear Differential Equations

$k$th order inhomogeneous linear ODE

They generally take the form of

$$a_k(x)\frac{d^{k}y}{d{x}^k}+a_{k-1}(x)\frac{d^{k-1}y}{d{x}^{k-1}}+\cdots +a_1(x)\frac{dy}{dx}+a_0(x)y=f(x)$$

where $a_k()x)\ne 0$

$k$th order homogeneous linear ODE

Same as above but with $f(x)=0$ i.e.

$$a_k(x)\frac{d^{k}y}{d{x}^k}+a_{k-1}(x)\frac{d^{k-1}y}{d{x}^{k-1}}+\cdots +a_1(x)\frac{dy}{dx}+a_0(x)y=0$$

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Solving First Order Linear Differential Equations

Suppose we have a first order differential equation ($k=1$)
Then we have the general form

$$\frac{dy}{dx}+p(x)y=q(x)$$

When $q(x)=0$, in other words in it’s homogenous form then it is separable

The inhomogeneous form can be solved using the integrating factor $I(x)$ by

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

Multiplying both sides by $I(x)$ then

$$e^{\int p(x)dx}\frac{dy}{dx}+p(x)e^{\int p(x)dx}y=e^{\int p(x)dx}q(x)$$

Using product rule then

$$\frac{d}{dx}(e^{\int p(x)dx}\cdot y)=e^{\int p(x)dx}q(x)$$

Hence we get

$$y=e^{-\int p(x)dx}\left[\int e^{\int p(x)dx}q(x)dx+c\right]$$

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