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Compute y(4) , the solution int = of the following ordinary differential equation d with y(0) =1 using the classical RK4 method with h = 0.5 Use 16 significant figu...

Question

Compute y(4) , the solution int = of the following ordinary differential equation d with y(0) =1 using the classical RK4 method with h = 0.5 Use 16 significant figures in your calculations and give your answer with at least significant figuresAnswer: 5.77933

Compute y(4) , the solution int = of the following ordinary differential equation d with y(0) =1 using the classical RK4 method with h = 0.5 Use 16 significant figures in your calculations and give your answer with at least significant figures Answer: 5.77933



Answers

Use Euler's method with the specified step size to determine the solution to the given initial-value problem at the specified point. $$y^{\prime}=4 y-1, \quad y(0)=1, \quad h=0.05, \quad y(0.5)$$.

Stay. We're gonna using the runner cutter method to approximate the differential equation Y prime is able to one minus y. To do this, we're gonna define full Constance Kay one k two k three and careful. Oh, it's rigidly defined with respect to one another. Now, Kate one is obtained by taking the right hand side of the different cell equation and plugging in. Why m que tu is obtained by taking the right hand side and plugging in Y N s K one over two. Okay, three by plugging in. Why end plus que tu over tube and kay full by plugging in y end plus K three note over to this time and then multiplying everything. Bye. Hate These are the values of K about the K values that you want for this differential equation. Now, the runner cut a method, then approximates the differential equation by the interest of values. Why one plus. Okay, one plus two K two plus two K three plus four or over six. Now. Um, except this one is also gonna be in committed by the step size. Um, now, in this case, hey, H, is there a 0.25 and we're gonna try and compute all the way X equals one. Now it's strongly recommend calculating this using a computer program in your favorite programming language. In this case, I use python, but you can also use it by do it by hand, but it's going to take quite a while. If we use a computer, will find that X here and y zero above zero x one. Is there a 10.25 and why one is 0.2211 x two is Europe in five? Why two is there a 0.3 93 x three is 0.75 Why three 0.5 to 76 and when exports warm. Finally, why Fourth will be equal to 0.632 which is very close to the Taylor approximation we calculated in the last problem.


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