Solve using both methods please Euler and Runge-KuttaConsider the following IVP:dyyexp(3x)dxy(0) = 1Develop an analytical solution to the IVP. Using a step size of 0.1, determine by handy(0.3) and compare with the analytical solution using:The explicit Euler methodThe 4th Order Runge-Kutta method

Name: Solve using both methods please Euler and Runge-KuttaConsider the foll
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Description: Solve using both methods please Euler and Runge-KuttaConsider the following IVP:dyyexp(3x)dxy(0) = 1Develop an analytical solution to the IVP. Using a step size of 0.1, determine by handy(0.3) and compare with the analytical solution using:The explic

We have dydx=ye3xdyy=e3xdxIntegrating both sides we have∫dyy=∫e3xdxlny=e3x3+ctherefore y=c ee3x3when…

Answered: Euler and Runge-Kutta Consider the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Solve using both methods please

Euler and Runge-Kutta
Consider the following IVP:
dy
yexp(3x)
dx
y(0) = 1
Develop an analytical solution to the IVP. Using a step size of 0.1, determine by hand
y(0.3) and compare with the analytical solution using:
The explicit Euler method
The 4th Order Runge-Kutta method

Expert Solution

Step 1

We have $\frac{d y}{d x} = y e^{3 x} \frac{d y}{y} = e^{3 x} d x I n t e g r a t i n g b o t h s i d e s w e h a v e \int \frac{d y}{y} = \int e^{3 x} d x \ln y = \frac{e^{3 x}}{3} + c t h e r e f o r e y = c e^{\frac{e^{3 x}}{3}} w h e n x = 0 a n d y = 1 c = e^{- \frac{e^{0}}{3}} = e^{\frac{- 1}{3}} t h e r e f o r e y = e^{\frac{e^{3 x} - 1}{3}}$

Step 2

(a) Euler Method

table for Euler approximation

where step size (h)=0.1

$y (0.3) \approx 1.475963 a n d y (0.3) = 1.626669 w i t h e r r o r = 0.150696$

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