First Order DE Solving Techniques

Ravinder Kumar, Sep 29, 2019

First order differential equation is the simplest of the differential equations. In this book we put together basic techniques used for solving first order differential equations.

Table of Contents

  1. Slope Field
    1. Slope Field
  2. Differential Equations With Separable Variables
    1. DE with Variables Separable
    2. Separable Variables DE Solver
  3. First Order Homogeneous Differential Equation
    1. First Order Homogeneous Differential Equation
    2. Homogeneous DE Solver
  4. Exact Differential Equation
    1. Exact Differential Equation
    2. Exact DE Solver
  5. Linear Differential Equation of First Order
    1. First Order Linear DE
    2. Linear Differential Equation Solver

1.

Slope Field

Slope field is visualization of the solution of a first order first degree differential equation. Such a differential equation can be written in the form [math]\frac{dy}{dx}=f(x,y)[/math] which means [math]f(x,y)[/math] is slope of the graph of the solution of the differential equation at the point [math](x,y)[/math].

  • 1. Slope Field

1.1.

Slope Field

This GeoGebra applet is intended to explore slope field of an ordinary differential equation. Slope field can be used to sketch graph of a solution of an ODE without solving the ODE. Instructions to use the applet are given in Graphic View 1.[br][br]For more on ODE check my online book [br]"Flipped Classroom Calculus of Single Variable"[br]https://versal.com/learn/vh45au/
Slope Field
– Ravinder Kumar

2.

Differential Equations With Separable Variables

A differential equation [math]M(x,y) dx+N(x,y) dy=0[/math] is said to be variables separable type if it can be rewritten in the form [math]f(x) dx+g(y) dy=0[/math].

  • 1. DE with Variables Separable
  • 2. Separable Variables DE Solver

2.1.

DE with Variables Separable

A DE M(x,y)dx+N(x,y)dy=0 is said to be of the type variables separable if it can be rewritten the form [br]f(x)dx+g(y)dy=0[br]The applet contains stepwise solution of an example and many randomly generated questions. The purpose of this applet is to facilitate practice on solving this type of DEs. Follow the instructions which can be displayed by checking the checkbox Instructions
– Ravinder Kumar

2.2.

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3.

First Order Homogeneous Differential Equation

A differential equation [math]M(x,y) dx+N(x,y) dy=0[/math] is said to be homogeneous if it can be written in the form [math]\frac{dy}{dx}=f(x,y)[/math], where [math]f(x,y)[/math] is homogeneous function of degree zero. Alternatively, such a differential equation can be written in the form [math]\frac{dy}{dx}=f(y/x)[/math]. A method to solve such a differential equation is to use substitution [math][y=ux/math] which converts the differential equation into one with separable variables.

  • 1. First Order Homogeneous Differential Equation
  • 2. Homogeneous DE Solver

3.1.

First Order Homogeneous Differential Equation

A first order differential equation is called homogeneous if it can be written in the form [math]\frac{dy}{dx}=f\left(\frac{y}{x}\right)[/math]. Its solution requires substitution [math]y=ux[/math], which converts it into a differential equation in u and x with variables separable. The purpose of this applet is to facilitate practice on solving this type of differential equation. A step-wise example is provided. Follow the instructions given in the applet.
– Ravinder Kumar

3.2.

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4.

Exact Differential Equation

A differential equation [math]M(x,y) dx+N(x,y) dy=0[/math] is said to be exact if [math\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}[ M/math].

  • 1. Exact Differential Equation
  • 2. Exact DE Solver

4.1.

Exact Differential Equation

The purpose of this applet is to facilitate practice on solving an exact differential equation. A first order first degree differential equation is said to be exact if, when written in the form M(x,y) dx+N(x,y) dy = 0,[br][math]M_y=N_x[/math]. The subsuffix denotes the variable for taking partial derivative. Follow the instructions on the applet.
– Ravinder Kumar

4.2.

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5.

Linear Differential Equation of First Order

A differential equation [math]M(x,y) dx+N(x,y) dy=0[/math] is said to be linear if it can be rewritten in the form [math]\frac{dy}{dx}+P(x) y=Q(x)[/math]. A method to solve such a differential equation is to use the integrating factor [math]e^{\int{P dx}}[/math].

  • 1. First Order Linear DE
  • 2. Linear Differential Equation Solver

5.1.

First Order Linear DE

A first [url=#]order[/url] linear differential equation can be written in the form[br][math]\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)[/math][br]A method to solve this kind of differential equation is to use integrating factor [math]e^{\int Pdx}[/math].[br]The purpose of this applet is to facilitate [url=#]practice[/url] solving these differential equations.[br]Follow instructions given on the applet. Some of the answers involve absolute values and sgn function because of the piecewise nature of integrand involved. One may use [math]\left|t\right|=t[/math] and [math]sgn\left(t\right)=1[/math] to compare the user answer without sgn function.
– Ravinder Kumar

5.2.

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