1. Find y(0.2) for `y'=x^2y-1`, y(0) = 1, with step length 0.1 using Taylor Series method

Solution:
Given `y'=x^2y-1, y(0)=1, h=0.1, y(0.2)=?`

Here, `x_0=0,y_0=1,h=0.1`

Differentiating successively, we get
`y'=x^2y-1`

`y''=2xy+x^2y'`

`y'''=2y+4xy'+x^2y''`

`y''''=6y'+6xy''+x^2y'''`

Now substituting, we get
`y_0'=x_0^2y_0-1=-1`

`y_0''=2x_0y_0+x_0^2y_0'=0`

`y_0'''=2y_0+4x_0y_0'+x_0^2y_0''=2`

`y_0''''=6y_0'+6x_0y_0''+x_0^2y_0'''=-6`

Putting these values in Taylor's Series, we have
`y_1 = y_0 + hy_0' + h^2/(2!) y_0'' + h^3/(3!) y_0''' + h^4/(4!) y_0'''' + ...`

`=1+0.1*(-1)+(0.1)^2/(2!)*(0)+(0.1)^3/(3!)*(2)+(0.1)^4/(4!)*(-6)+...`

`=1-0.1+0+0.00033+0+...`

`=0.90031`

`:.y(0.1)=0.90031`

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Again taking `(x_1,y_1)` in place of `(x_0,y_0)` and repeat the process

Now substituting, we get
`y_1'=x_1^2y_1-1=-0.991`

`y_1''=2x_1y_1+x_1^2y_1'=0.17015`

`y_1'''=2y_1+4x_1y_1'+x_1^2y_1''=1.40592`

`y_1''''=6y_1'+6x_1y_1''+x_1^2y_1'''=-5.82983`

Putting these values in Taylor's Series, we have
`y_2 = y_1 + hy_1' + h^2/(2!) y_1'' + h^3/(3!) y_1''' + h^4/(4!) y_1'''' + ...`

`=0.90031+0.1*(-0.991)+(0.1)^2/(2!)*(0.17015)+(0.1)^3/(3!)*(1.40592)+(0.1)^4/(4!)*(-5.82983)+...`

`=0.90031-0.0991+0.00085+0.00023+0+...`

`=0.80227`

`:.y(0.2)=0.80227`