Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 03

MATH102 December 2013

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[hide] Question A 03
Consider the differential equation and initial condition ${\frac {dy}{dt}}=2-y^{2},\quad y(0)=1$ Use Euler's method with one step of size $\Delta t=0.1$ to approximate the value of the solution at time $t=0.1$ (a) $y(0.1)=2$ (b) $y(0.1)=2.1$ (c) $y(0.1)=2.2$ (d) $y(0.1)=1.2$ (e) $y(0.1)=1.1$

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[show] Hint
Euler's method is an iterative method whereby we start with the initial value, plug it into our differential equation to find the slope, then use the slope of this tangent line to find the next point on the curve that is $\Delta t$ units away.

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[show] Solution
We start at the point $(0,1)$ , where the slope of the tangent line at this point is ${\frac {dy}{dx}}=2-y^{2}=2-[y(0)]^{2}=2-1=1$ By applying Euler's method with a step size of $\Delta t=0.1$ , our next point would be at the value $y(0.1)=y(0)+\left.{\frac {dy}{dx}}\right\|_{x=0}(0.1-0)=1+0.1=1.1$ , which is answer ${\color {blue}{\text{(E)}}}$ .