We show particular techniques to solve particular types of first order differential equations. The techniques were developed in the eighteenth and nineteenth centuries and the equations include linear equations, separable equations, Euler homogeneous equations, and exact equations. This way of studying differential equations reached a dead end pretty soon. Most of the differential equations cannot be solved by any of the techniques mentioned in the first sections. Instead of solving the equations they tried to show whether an equation has solutions or not, and what properties such solution may have. This is less information than obtaining the solution, but it is still valuable information. We present theorems describing the existence and uniqueness of solutions to a wide class of first order differential equations.

Linear Constant Coefficient Equations

For example, let us mention Newton’s and Lagrange’s equations for classical mechanics, Maxwell’s equations for classical electromagnetism, Schrödinger equation for quantum mechanics, and Einstein’s equation for the general theory of gravitation.

Let us look at few real-life examples:

(a) Radioactive Decay: The amount u of a radioactive material changes in time as follows,

$\frac{\text{d}u}{\text{d}t}\left(\begin{array}{c}t\end{array}\right)= -ku\left(t\right),$                               $k>0,$

where k is a positive constant representing radioactive properties of the material.

This is a first order Ordinary Differential Equation (ODE).

(b) Newton’s law: Mass times acceleration equals force, $ma= f$ where $m$ is the particle mass,

$m\frac{\text{d}^{2}x}{\text{d}t^{2}}(t)= f\left(t,x(t),\frac{\text{d}x}{\text{d}t}(t)\right),$

where the unknown is $x(t)$—the position of the particle in space at the time $t$. As we see above, the force may depend on time, on the particle position in space, and on the particle velocity.

This is a second order Ordinary Differential Equation (ODE).

In both the examples mentioned above, we notice that the unknown function is dependent on a single Independent Variable. This type of equation is called as “Ordinary Differential Equation“.

We know, that the order of a differential equation is the highest derivative order that appears in the equation. Hence, the Time decay equation in example (a) is first order and Newton’s equation in example (b) is second order.

Linear Differential Equation

A first order ODE on the unknown y is    $y'(t)= f(t,y(t)),$

where $f$ is given and $y' = \frac {dy}{dt}$.

The equation is linear if the source function $f$ is linear on its second argument, $y' = a(t)y + b(t)$.

The linear equation has constant coefficients (as shown in example a) if both a and b above are constants. Otherwise the equation has variable coefficients (as shown in example b). Let’s discuss few examples.

Examples:

a. $y'=3y+5$

b. $y'=-\frac{\text{2}}{\text t}y+5t$

Let’s solve a linear differential equation problem,

Problem 1:  $y' = 2y +3$

Let’s integrate on both sides with respect to $t$,

$\int_{}^{} y'(t)dt = 2\int_{}^{} y(t)dt+3t+c,$                  $c\in R$

According to the Theorem of Calculus, $y(t)=\int_{}^{}y'(t)dt$

so we get, $y(t)=2\int_{}^{} y(t)dt+3t+c$.

Integrating both sides of the differential equation is not enough to find a solution y. We still need to find a primitive of y. We have only rewritten the original differential equation as an integral equation. Simply integrating both sides of a linear equation does not solve the equation.
We now state a precise formula for the solutions of constant coefficient linear equations. The proof relies on a new idea—a clever use of the chain rule for derivatives.

THEOREM of Constant Coefficients:

The linear differential equation, $y'=ay+b$, where $a\neq0,b$ constants has infinitely many solutions,

$y(t)=ce^{at}-\frac{b}{a}$,                           $c \in R$.

This is the general solution of the differential equation $y'=ay+b$.

Theorem says that the equation $y'=ay+b$ has infinitely many solutions, one solution for each
value of the constant c, which is not determined by the equation.

The Integrating Factor Method

The Theorem of Constant Coefficients cannot be generalized in a simple way to all linear equations with variable coefficients. However, there is a way to solve linear equations with both constant and variable coefficients—the Integrating Factor Method.

The step-by-step approach for Solving the numerical is as follows:

Step 1: Make sure your Linear first order ODE is arranged in the following form.

$\frac{\text{d}y}{\text{d}x}+P(x)y=Q(x)$,            where $P(x)$ and $Q(x)$ can be functions of x or constants.

Step 2: Evaluate the value of Integrating Factor by using the formula, $IF = e^{\int_{}^{}P(x)dx}$.

Step 3: Multiply the equation obtained in Step 1 by $IF$ as shown below.

$IF.\frac{\text{d}y}{\text{d}x}+IF.P(x)y=IF.Q(x)$.

If you notice carefully here, the LHS part of this equation represents the Product Rule. Thus, the LHS part can be modified as $IF.\frac{\text{d}y}{\text{d}x}+IF.P(x)y=\frac{\text{d}(IF.y)}{\text{d}x}$

Step 4: Integrate both sides with respect to x.

$IF.y = \int_{}^{}IF.Q(x)dx$

Step 5: Divide both sides of the obtained equation by $IF$.

The Initial Value Problem

Sometimes in physics one is not interested in all solutions to a differential equation, but only in those solutions satisfying extra conditions. For example, in the case of Newton’s second law of motion for a point particle, one could be interested only in solutions such that the particle is at a specific position at the initial time. Such condition is called an initial condition, and it selects a subset of solutions of the differential equation. An initial value problem means to find a solution to both a differential equation and an initial condition.

Linear Variable Coefficient Equations

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Separable Equations

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Exact Differential Equations

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Applications of Linear Equations

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Non-Linear Equations

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