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Numerade Educator

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Problem 22 Medium Difficulty

Use Euler's method with step size 0.2 to estimate $y(1)$ where $y(x)$ is the solution of the initial-value problem $y^{\prime}=x^{2} y-\frac{1}{2} y^{2}, y(0)=1$

Answer

$y(1) \approx 0.818347$

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Video Transcript

were given initial value problem. We were asked to use Oilers method to approximate the value of the solution. At a given value of X, we're given the differential equation y prime goes X squared y minus 1/2 y squared with initial value. Why it zero equals one. And the steps eyes were asked to use his each equals point to you were asked to find the value of why of one. Our initial value tells us that X zero is equal to zero and y zero. His wife of X zero, which is one x one is x zero plus er step size, which is 00.2 and why one is equal to why zero, which is one plus our step size point to times the value of our function. X squared y minus 1/2 y squared at 01 just simply negative. 1/2 1 squared is equal to one plus point to times negative. 0.5 is equal to point 0.9. X two is going to be x one, plus her step function by step just point for And why to is why one 0.9 plus step size 22 times Value of our function at 0.2 point nine. This is 0.2 squared times 0.9 minus 1/2 times 0.9 squared you point 82 six to x three is going to be X two plus or step size. So 20.6 in why three it's going to be. Instead of writing out why? To simply write Why, too, Plus our steps. Signs point to terms of a ver functions at point for why to 4.4 squared y two minus 1/2. Why two squared? This is too 0.78 for three 778 Approximately X four is equal to 0.6 plus or step size or 0.8. And why four? It's going to be equal to buy three plus our steps eyes point to times are function evaluated x three y three, which is 0.6 Swear times Wife three minus 1/2. Why three squared, which is equal to 0.779 32 81 Approximately. Finally, we have that X five is going to be X for pleasure. Step size, which is one in Y five, is equal to y for plus step size 0.2 times or function evaluated at X for y for just point it squared. Why four minus one? Have why four squared? Which is equal to approximately point 81 83 469 And we have that y five. This is approximately equal to why have one. So our answer is going 8183 for 69 or 0.81 83 for seven after rounding.