1. Consider the IVP y' (t) = 5 + 4t y(t) , y(0) = 1. 0...
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1. Consider the IVP y' (t) = 5 + 4t y(t) , y(0) = 1. 0 t 4, a. (2 points) Use Wolfram Alpha to obtain a function y that satisfies the differential equation and the intiial condition. Include a screenshot of the results in your write-up. b. Perform the following in MATLAB: i. (3 points) Plot the solution obtained in part (a) over t = [0, 4]. ii. (3 points) Make use of the provided RK4 function in RK4. m to obtain iterates at N = 20 equidistant points, starting at 0 and ending at 4. Give a scatter plot of the iterates on top of the previous plot. function unext = RK 4 (f, h, t, u) k1 = h*f (t, u): k2 = h*f (t+h/2, u+k1/2): k3 =h*f (t+h/2, u+k2/2); h*f (t+h, utk 3); k4 = k4 h*f (t+h, u+k3); unext=u+(k1+2*k2+2*k3+k4)/6: end = Let U CRX R be an open set containing [a, b] x R where a b. Let f: U R and yo E R. Recall that the IVP is to find a continuously differentiable function y : [a, b] R satisfying y(a) = yo, y'(t) = f(t, y(t)) a Theorem 11.OIE. The IVP is equivalent to the integral equation y(t)- yo = [*(7,y(7))dr, t[a,b]. The integral equation is often written in the form y(t) = yo + + [ f(r, y(T)dT. a 1. Consider the IVP y' (t) = 5 + 4t y(t) , y(0) = 1. 0 t 4, a. (2 points) Use Wolfram Alpha to obtain a function y that satisfies the differential equation and the intiial condition. Include a screenshot of the results in your write-up. b. Perform the following in MATLAB: i. (3 points) Plot the solution obtained in part (a) over t = [0, 4]. ii. (3 points) Make use of the provided RK4 function in RK4. m to obtain iterates at N = 20 equidistant points, starting at 0 and ending at 4. Give a scatter plot of the iterates on top of the previous plot. function unext = RK 4 (f, h, t, u) k1 = h*f (t, u): k2 = h*f (t+h/2, u+k1/2): k3 =h*f (t+h/2, u+k2/2); h*f (t+h, utk 3); k4 = k4 h*f (t+h, u+k3); unext=u+(k1+2*k2+2*k3+k4)/6: end = Let U CRX R be an open set containing [a, b] x R where a b. Let f: U R and yo E R. Recall that the IVP is to find a continuously differentiable function y : [a, b] R satisfying y(a) = yo, y'(t) = f(t, y(t)) a Theorem 11.OIE. The IVP is equivalent to the integral equation y(t)- yo = [*(7,y(7))dr, t[a,b]. The integral equation is often written in the form y(t) = yo + + [ f(r, y(T)dT. a
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