Please help with this math question

Question

1. . The ordinary differential equation :

dy/dx = e^(-x^2) , y(0)=1 ; maybe solved using

a. No technique covered in class

b. Integration by parts

c. The substitution y = xv

d. Euler method

Answer 1

The function g(x) = e ^( -x² ) is continuous on ( -∞ ,∞ ) but its antiderivative is not an elementary function.

Letting t as a dummy variable of integration , we can write

∫₀^x [ d y / d t ] d t = ∫₀^x [ e ^( -t² ) ] d t

y ( t ) | ₀^x = ∫₀^x [ e ^( -t² ) ] d t

y (x) - y( 0 ) = ∫₀^x [ e ^( -t² ) ] d t

y ( x ) - 1 = ∫₀^x [ e ^( -t² ) ] d t

y ( x ) = 1 + ∫₀^x [ e ^( -t² ) ] d t