Euler's Method

Calculus Level 2

If y ( x ) y(x) satisfies the differential equation d y d x = x 2 + x y \frac{dy}{dx}=x^2+xy and y ( 0 ) = 2 y(0)=2 , what is the approximate value of y ( 4 ) y(4) , using Euler's method with step size 1?


The answer is 85.

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1 solution

Samir Khan
Jun 11, 2016

We have ( x 0 , y 0 (x_0,y_0 =(0,2)), and then: ( x 1 , y 1 ) = ( 0 + 1 , 2 + 1 ( 0 2 + 0 2 ) ) = ( 1 , 2 ) , ( x 2 , y 2 ) = ( 1 + 1 , 2 + 1 ( 1 2 + 1 2 ) ) = ( 2 , 5 ) , ( x 3 , y 3 ) = ( 2 + 1 , 5 + 1 ( 2 2 + 2 5 ) ) = ( 3 , 19 ) , ( x 4 , y 4 ) = ( 3 + 1 , 19 + 1 ( 3 2 + 3 19 ) ) = ( 4 , 85 ) , \begin{aligned} (x_1,y_1)&=(0+1,2+1\cdot (0^2+0\cdot 2))=(1,2),\\ (x_2,y_2)&=(1+1,2+1\cdot (1^2+1\cdot 2))=(2,5),\\ (x_3,y_3)&=(2+1,5+1\cdot (2^2+2\cdot 5))=(3,19),\\ (x_4,y_4)&=(3+1,19+1\cdot (3^2+3\cdot 19))=(4,85), \end{aligned}

so the answer is 85.

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