Solve numerical differential equation using Euler method (2nd order derivative) calculator

SolutionHelpInput functions

Solution

Solution provided by AtoZmath.com

1. Find y(1) for `y''=-4z-4y`, `x_0=0, y_0=0, z_0=1`, with step length 0.1

2. Find y(0.1) for `y''=1+2xy-x^2z`, `x_0=0, y_0=1, z_0=0`, with step length 0.1

3. Find y(0.2) for `y''=xz^2-y^2`, `x_0=0, y_0=1, z_0=0`, with step length 0.2

Example

1. Find y(0.1) for `y''=1+2xy-x^2z`, `x_0=0, y_0=1, z_0=0`, with step length 0.1 using Euler method (2nd order derivative)

Solution:
Given `y^('')=1+2xy-x^2z, y(0)=1, y'(0)=0, h=0.1, y(0.1)=?`

put `(dy)/(dx)=z` and differentiate w.r.t. x, we obtain `(d^2y)/(dx^2)=(dz)/(dx)`

We have system of equations
`(dy)/(dx)=z=f(x,y,z)`

`(dz)/(dx)=1+2xy-x^2z=g(x,y,z)`

Euler method for second order differential equation
`y_1=y_0+hf(x_0,y_0,z_0)=1+(0.1)*f(0,1,0)=1+(0.1)*(0)=1+(0)=1`

`:.y(0.1)=1`

`n`	`x_n`	`y_n`	`z_n`	`x_(n+1)`	`y_(n+1)`	`z_(n+1)`
0	0	1	0	0.1	1

Input functions

Sr No.	Function	Input value
1.	`x^3`	x^3
2.	`sqrt(x)`	sqrt(x)
3.	`root(3)(x)`	root(3,x)
4.	sin(x)	sin(x)
5.	cos(x)	cos(x)
6.	tan(x)	tan(x)
7.	sec(x)	sec(x)
8.	cosec(x)	csc(x)
9.	cot(x)	cot(x)
10.	`sin^(-1)(x)`	asin(x)
11.	`cos^(-1)(x)`	acos(x)
12.	`tan^(-1)(x)`	atan(x)
13.	`sin^2(x)`	sin^2(x)
14.	`log_y(x)`	log(y,x)
15.	`log_10(x)`	log(x)
16.	`log_e(x)`	ln(x)
17.	`e^x`	exp(x) or e^x
18.	`e^(2x)`	exp(2x) or e^(2x)
19.	`oo`	inf