Describe how the fourth-order Runge-Kutta method can be used to produce a table of values for the function

 

$$f(x)=\int_0^x e^{-t^2}\ \mathsf dx$$

 

at $100$ equally spaced points in the unit interval.

Answer: Find an appropriate initial-value problem whose solution is f. Solve df =e^{−x^2}, f(0)=0.

The answer is a little confusing and can anyone write out the full answer? I suppose my question is how after i take the derivative and transform the intergral equation into a differential equation with initial value, how do i know the step size h? n how do i know the initial value? I know what runge kutta 4th order is The runge kutta method is

$$k_1 = f(t_0,x(t_0))$$ $$k_2 = f(t_0+h/2,x(t_0)+k_1/2)$$ $$k_3 = f(t_0+h/2,x(t_0)hk_2/2)$$ $$k_4 = f(t_0+h,x(t_0)+k_3)$$ Then $$x(t_0+h)= x(t_0) + h\frac{k_1+2k_2+2k_3+k_4}{6}$$

Describe how the fourth-order Runge-Kutta method can be used to produce a table of values for the function

$$f(x)=\int_0^x e^{-t^2}\ \mathsf dx$$

at $100$ equally spaced points in the unit interval.

Answer: Find an appropriate initial-value problem whose solution is f. Solve df =e^{−x^2}, f(0)=0.

The answer is a little confusing and can anyone write out the full answer? I suppose my question is how after i take the derivative and transform the intergral equation into a differential equation with initial value, how do i know the step size h? n how do i know the initial value? I know what runge kutta 4th order is The runge kutta method is

$$k_1 = f(t_0,x(t_0))$$ $$k_2 = f(t_0+h/2,x(t_0)+k_1/2)$$ $$k_3 = f(t_0+h/2,x(t_0)hk_2/2)$$ $$k_4 = f(t_0+h,x(t_0)+k_3)$$ Then $$x(t_0+h)= x(t_0) + h\frac{k_1+2k_2+2k_3+k_4}{6}$$

Describe how the fourth-order Runge-Kutta method can be used to produce a table of values for the function

$$f(x)=\int_0^x e^{-t^2}\ \mathsf dx$$

at $100$ equally spaced points in the unit interval.

Answer: Find an appropriate initial-value problem whose solution is f. Solve df =e^{−x^2}, f(0)=0.

The answer is a little confusing and can anyone write out the full answer? I suppose my question is how you can take the derivative and transform the intergral equation into a differential equation with initial value? I know what runge kutta 4th order is The runge kutta method is

$$k_1 = f(t_0,x(t_0))$$ $$k_2 = f(t_0+h/2,x(t_0)+k_1/2)$$ $$k_3 = f(t_0+h/2,x(t_0)hk_2/2)$$ $$k_4 = f(t_0+h,x(t_0)+k_3)$$ Then $$x(t_0+h)= x(t_0) + h\frac{k_1+2k_2+2k_3+k_4}{6}$$

Describe how the fourth-order Runge-Kutta method can be used to produce a table of values for the function

$$f(x)=\int_0^x e^{-t^2}\ \mathsf dx$$

at $100$ equally spaced points in the unit interval.

Answer: Find an appropriate initial-value problem whose solution is f. Solve df =e^{−x^2}, f(0)=0.

The answer is a little confusing and can anyone write out the full answer? I suppose my question is how after i take the derivative and transform the intergral equation into a differential equation with initial value, how do i know the step size h? n how do i know the initial value? I know what runge kutta 4th order is The runge kutta method is

$$k_1 = f(t_0,x(t_0))$$ $$k_2 = f(t_0+h/2,x(t_0)+k_1/2)$$ $$k_3 = f(t_0+h/2,x(t_0)hk_2/2)$$ $$k_4 = f(t_0+h,x(t_0)+k_3)$$ Then $$x(t_0+h)= x(t_0) + h\frac{k_1+2k_2+2k_3+k_4}{6}$$