Taylor Series Method To Solve First Order Differential Equations (Numerical Solution)

Summary

In this educational video, the host, Suy, introduces viewers to solving differential equations using the Taylor series method. The script explains the process step by step, starting with basic differentiation and progressing to higher orders. Suy demonstrates how to calculate the numerical solution for a given first-order differential equation with initial conditions, using the Taylor series formula. The video also mentions alternative methods like Euler's method and Milne's predictor-corrector method, with links provided in the description. The host encourages viewers to subscribe for more informative content.

Takeaways

• 📚 The video is an educational tutorial by Suy on solving differential equations using the Taylor series method.
• 🔢 The script introduces three methods for solving differential equations: Euler's method, Milne's predictor-corrector method, and Adams-Bashforth predictor-corrector method, with links to related videos provided in the description.
• 📈 The primary focus is on the Taylor series method, which is explained in a simple language suitable for understanding the numerical solution of differential equations.
• 📘 The given differential equation to solve is y' = 1 + xy with the initial condition y(0) = 1.
• 📝 The script explains the notation y' as the first derivative of y with respect to x, and the initial conditions as x_0 and y_0.
• 🔍 The process involves differentiating the given differential equation repeatedly to obtain higher-order derivatives.
• 📌 The increment in x is denoted by h, and in this case, h = 0.1.
• 🔢 The script demonstrates the calculation of y_0, y_1, and y_2 using the Taylor series formula, which includes terms up to the third derivative.
• 📈 The Taylor series formula used is y_n = y_0 + h/factorial(1) · y'_0 + h^2/factorial(2) · y''_0 + h^3/factorial(3) · y'''_0.
• 📊 The video provides step-by-step calculations to find the numerical solutions for y at x = 0.1 and x = 2.
• 👍 The presenter encourages viewers to like, share, and subscribe to the YouTube channel for more educational content.

Q & A

What is the main topic of the video?

The main topic of the video is how to find the numerical solution of differential equations using the Taylor series method.

What are the alternative methods mentioned for solving differential equations?

The alternative methods mentioned are the Euler's method, Milne's predictor-corrector method, and Adams-Bashforth predictor-corrector method.

Where can viewers find more information about the alternative methods?

Viewers can find more information about the alternative methods in the video description where links to separate videos on each method are provided.

What does 'y' with an apostrophe (') represent in the context of the video?

In the context of the video, 'y' with an apostrophe (') represents the first derivative of y with respect to x, which is dy/dx.

What is the given differential equation in the video?

The given differential equation in the video is dy/dx = 1 + xy, with the initial condition y(0) = 1.

What is the purpose of the increment 'h' in the Taylor series method?

The increment 'h' in the Taylor series method is the step size used to approximate the solution at a new point in the x-direction.

How is the first derivative 'y0' calculated in the video?

The first derivative 'y0' is calculated using the formula 1 + x0 * y0, where x0 and y0 are the initial conditions.

What is the formula used to calculate the second derivative 'y''?

The second derivative 'y'' is calculated using the formula y + x * y', where y' is the first derivative.

How is the Taylor series formula structured in the video?

The Taylor series formula in the video is structured as y1 = y0 + h/factorial(1) * y'0 + h^2/factorial(2) * y''0 + h^3/factorial(3) * y'''0, where the number of terms corresponds to the order of the derivative.

What are the final numerical solutions for y(0.1) and y(2) obtained in the video?

The final numerical solutions obtained in the video are y(0.1) = 1.15 and y(2) = 1.222.

How can viewers stay updated with the channel's new content?

Viewers can stay updated with the channel's new content by subscribing to the YouTube channel and turning on notifications.

Outlines

00:00

📚 Introduction to Solving Differential Equations with Taylor Series

In this video, the host, Suy, introduces the audience to the method of solving differential equations using Taylor series. The script begins by mentioning alternative methods such as Euler's method, Milne's predictor-corrector method, and Adams-Bashforth predictor-corrector method, with links to these methods provided in the video description. The focus then shifts to the Taylor series method, starting with basic definitions like first-order differentiation (Dy/Dx) and initial conditions (y(0) = 1). Suy proceeds to demonstrate the process of differentiating the given differential equation step by step, explaining the significance of each step in the calculation. The video aims to simplify the complex topic and make it accessible to viewers.

05:04

🔍 Detailed Walkthrough of the Taylor Series Method Application

The second paragraph delves into the application of the Taylor series method to find the numerical solution of the given differential equation at specific points. Suy illustrates the iterative process of calculating the derivative values (y', y'', etc.) using the initial conditions and the increment in x (h = 0.1). The explanation includes the use of the Taylor series formula to approximate the value of y at x = 0.1 and x = 2, with a step-by-step breakdown of the calculations. Suy emphasizes the importance of using the correct factorial values and the relationship between the number of derivatives and the factorials in the formula. The paragraph concludes with the final numerical solutions for y at the specified points, providing a clear example of how to apply the Taylor series method to differential equations.

Keywords

💡Differential Equations

Differential equations are equations that relate a function to its derivatives. In the context of the video, these equations are used to model various phenomena in science and engineering. The video focuses on solving first-order differential equations, specifically the given equation y' = 1 + xy, where y' denotes the first derivative of y with respect to x.

💡Taylor Series Method

The Taylor series method is a mathematical technique used to approximate the solution of differential equations by expanding the function in a series around a given point. The video script describes how to use this method to find numerical solutions for the given differential equation, starting from an initial condition and iteratively applying the Taylor series formula.

💡Euler's Method

Euler's method is a numerical technique for solving differential equations with a given initial value. It is one of the simplest methods for such problems. The video mentions this method as one of the alternatives to the Taylor series method for solving differential equations, indicating the variety of approaches available.

💡Milne's Predictor-Corrector Method

Milne's predictor-corrector method is another numerical technique for solving differential equations. It is an improvement over Euler's method, providing more accurate solutions. The video script lists this method among the alternatives to the Taylor series method, showcasing the different strategies one can use to solve such equations.

💡Adams-Bashforth Predictor-Corrector Method

The Adams-Bashforth predictor-corrector method is a numerical algorithm used to solve ordinary differential equations. It is a predictor-corrector method that offers better accuracy than single-step methods like Euler's. The video script briefly mentions this method, positioning it as one of several options for solving the given problem.

💡First Order Differentiation

First order differentiation refers to the process of finding the first derivative of a function. In the video, this concept is fundamental as it involves differentiating the given differential equation to apply the Taylor series method. The script explains how to differentiate the equation y' = 1 + xy to obtain higher-order derivatives.

💡Initial Condition

An initial condition is a specified value or set of values used to determine the unique solution to a differential equation. In the video, the initial condition is given as y(0) = 1, which is crucial for starting the numerical solution process using the Taylor series method.

💡Increment (H)

In numerical methods, the increment (often denoted as h) represents the step size used in the iterative process to approximate the solution. The video script uses h = 0.1 to increment the value of x from 0 to 0.1 and then to 0.2, which is essential for calculating the successive values of y using the Taylor series formula.

💡Taylor Series Formula

The Taylor series formula is a mathematical expression used to approximate a function near a given point. In the video, the formula is used to calculate the value of y at successive points based on the derivatives obtained from the differential equation. The script provides a step-by-step application of this formula using the calculated derivatives and the increment h.

💡Numerical Solution

A numerical solution is an approximate value obtained by applying numerical methods to solve mathematical problems, such as differential equations. The video's main theme revolves around finding numerical solutions for the given differential equation using the Taylor series method, with the script detailing the process of calculating y at x = 0.1 and x = 0.2.

Highlights

Introduction to solving differential equations using Taylor series method.

Mention of alternative methods: Euler's method, Milne's predictor-corrector method, and Adam-Bashforth predictor-corrector method.

Link to videos on alternative methods provided in the video description.

Explanation of the first-order differentiation notation y' = dy/dx.

Clarification of initial conditions with x0 and y0 values.

Increment in x value, denoted as h, is introduced with an example of h = 0.1.

Derivation of the differential equation y' = 1 + xy to find y'' and y'''.

Calculation of y0 using the first formula 1 + x0 * y0.

Use of the second formula to calculate y0 with y0 + x0 * y0.

Application of the third formula 2y0 + x0 * y0 to find y0.

Introduction of the Taylor series formula for approximating y1.

Calculation of y1 using the Taylor series formula with given values of y0 and h.

Explanation of the process for the second iteration to find y2.

Calculation of y-1 using the first formula with updated x1 and y1 values.

Use of the second formula to calculate y-1 with updated y1 and x1 values.

Application of the third formula to find y-1 with updated values.

Final calculation of y2 using the Taylor series formula with updated y-1 values.

Conclusion of the video with the obtained values for y at 0.1 and 2.

Call to action for viewers to subscribe and engage with the content.

Transcripts

hello my dear friends I am suy and in

this video I will tell you how to find

out numerical solution of differential

equations by tayor series method there

are more than one ways to solve these

type of questions number one is the

Oilers method number two is millness

predictor corrector method number three

is Adam bford predictor corrector method

I have videos on all those three methods

the link to them is given in the video

description below so for now I will tell

you how to solve it by Tor series method

in very easy language so let's start our

question is find y of 0.1 and Y of 2

from y Das = to 1 + XY where y of 0

equal to 1 so first some Basics this y

Das means Dy by DX that is first order

differentiation and this y of 0 = to 1

means the value inside the bracket

represents the X value and value at the

right hand side depents the Y value so

for the first sample it's called x0 and

y0 values next X1 y1 values and so on so

let's proceed to the solution given y

Das = 1 + x y next if we differentiate

it with respect to X you will get y Das

that is d y by DX s that is second order

differentiation equal to y + x into y

Das differentiating again you will get

yle Das that's equals to 2 y + x into y

Das that means in each step we'll use

previous steps value that's why it will

become more accurate x0 = to 0 from here

and y0 = to 1 from here and H = to 0.1 H

is the increment in x value first x

value is zero next point1 next Point 2

so that is Inc this by 0.1 now we'll

calculate the y-0 value our first set of

values and for y-0 Value we'll use x0

and y0 value so y- 0 equals to using the

first Formula 1 + x0 into y0 so that is

1 + 0 into 1 = to 1 next y- 0 using the

second formula y0 + x0 into y- 0 that is

1 + 0 into this value that is 1 next

y-0 that's equals to using this Formula

2 y0 +

x0 into y- 0 using the second value so

it will become 2 into 1 + 0 into 1

that's equal to 2 remember we are using

this expression because this is our

question in your question the expression

may be different but the process is same

just different differentiate

continuously and get the formula and

then using the x0 y0 value you will get

all the y0 Das values so next we'll use

the tailor series formula which is y1 =

y0 + h / factorial of 1 into y- 0 + h²

by factori of 2 into y-0 plus h CU by

factori of 3 into y

triple-0 you see the number of dashes is

equals to number below or the factorial

number here power is two factorial is

two and Das is two it's very easy to

remember now we'll put the values y0 is

1 H is .1 / factorial of 1 into y- 0

that is 1 plus h² by factorial of 2 into

y0 + H Cub by factorial of 3 which is 6

into y- 0 so you will get 1.10 53 that

is y of .1 = to

1.15 now for the second iteration our y

value will Y2 and all the Y values will

be incremented by 1 so previously it was

y Z's now they are all y1s so Y2 = to y1

+ H into factor of 1 into

y-1 a² by factor of 2 into

y-1 HQ fact of 3 into

y-1 and so on next we will calculate the

y-1 value using X1 and y1 values so

using the first Formula 1 + X1 into y1 =

to 1 + .1 into our obtained value that

is 1.1 05 which is 1.11

05 next

y-1 = y1 + X1 into y-1 y1 is 1.1 05 + X1

is .1 into y-1 value about this value

which is equal to

1.26 next

y-1 = 2 into y1 + X1 into

y-1 2 into

1.15 + X1 is 0.1 into y-1 our previous

value which is = to 2.

3316 now we'll put the obtained values

in Tor formula so you will get so y1 is

this H by

.1 into y-1 this Value Plus h² by

factorial of 2 into y-1 which is this

Value Plus H CU by factoral of 3 into

y-1 that is this value so we'll get 1.

222 so y of point2 = to 1. 222 so that's

it we have got our required values for y

of 0.1 and2 so this was my video on Tor

series how was the video let me know in

the comments below I will upload more

videos like this so don't forget to

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