Question

Use Taylor series method to find a solution of `dy/dx=xy+1,y(0)=0` X=0.2 taking h=0.1 correct upto 4 decimal places. 

Answer 1

(I)` dy/dx=xy+1 ,`         `x_0=, y_0=0=0,  h=0.1` 

`f(x,y)=1+xy` 

`y'=1+xy`                           ` y'0=1`  

`y''=xy+y`                          `y''0=0` 

`y'''=xy+2y'`                     ` y'''o=2` 

Taylor’s series is given by , 

`y(0.1)=y_0+h. y'_0+h^2/(2!) y''_0+h^3/(3!) y'''0 +` 

=`0+0.1(1)+0+ ((0.1)^3)/6`    (𝟐) 

`y(0.1)= 0.1003` 

(II) `x_1=0, y_1=0.1003,h=0.1` 

`y'=1+xy`                           ` y'0=1.01003` 

`y''=xy'+y`                        `y''0=0.201303`

`y'''=xy''+2y'`                `y'''0=2.0401903` 

∴` y(0.2)=0.1003+1.01003(0.1)+0.1^2/(2!) (0.201303)+0.1^3/6(2.0401903)` 

∴` y (0.2)=0202708`