Science:Math Exam Resources/Courses/MATH102/December 2014/Question B 05

MATH102 December 2014

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Question B 05
Consider the differential equation ${\dfrac {dy}{dt}}=2(1-y)$ with the initial condition $y(0)=2$. Using Euler’s method with a single step of size $h=\Delta t=1/4$, estimate the value of the solution at $t=1/4$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
The Euler’s method has form $y_{n+1}=y_{n}+\Delta tf(y_{n}),$ where $f(y)$ is the derivative of $y$, i. e. ${\frac {dy}{dt}}=f(y)$.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. To approximate $y(1/4)$, we use the Euler’s method $y_{n+1}=y_{n}+\Delta tf(y_{n}).$ with a single step $\Delta t=1/4$ and $n=0$, where ${\frac {dy}{dt}}=f(y)$. As $y_{0}=y(0)=2$, we have ${\begin{aligned}y(1/4)=y_{1}&=y_{0}+\Delta t\cdot 2(1-y_{0})\\&=2+{\frac {1}{4}}\cdot 2(1-2)\\&={\frac {3}{2}}.\end{aligned}}$

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