$$y^{\prime}=\sin y+e^{x} ; \quad y(0)=0$$

Answer

$$
y=x+x^{2}+\frac{x^{3}}{2}
$$

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Calculus 1 / AB

Fundamentals of Differential Equations

Chapter 8

Series Solutions of Differential Equations

Section 1

Introduction: The Taylor Polynomial Approximation

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Video Transcript

we want to find first renounce your terms in the terrorist serious problem approximation for the given initial. Well, the problem and initial l. A problem s why prime equals sign off acts plus white with initial conditions. Why of zero lost to zero. So they're falling on approximation for solution of this problem will be in the foreign form. It is some I from zero to end eyes during which is why at 0.0 derided by I factorial dams export I. So the first term here is wives your will ready? Annoyed It is zero. Now we need to compute a few derivatives and zero and we need it for three months. Year terms is in the sip Siris. So why private zero? We're gonna use ah already here to compute it. It is sign off zero loss while zero So this sign of zero So very close to zero. So now the second derivative Warren double prime. We can find it by different training's a sign off exports white it becomes sign X plus y prime were chance costs x plus y times one plus y Prime general here then second Judy fed zero camps Asano zero Dennis one, which is one Let's go to the next page and continue. So far, we found on the one and zero chill here. Now we need to find third derivative at find surgery two different shades that intuitive. So why Triple prime equals if two years of product ruined general here minus sign off acts plus wine. Thomas one loss Weiss Court That's very white Prime squid Applaud the second terms of product release. Sign off X plus y time. Second Jewish for why So now we have a lead surge. Useful? Why at zero. So here we have a minus sign 00 times one plus y prime or zero squared plus sign of zero time second usual wide zero. The first term will be zero because sign was, you know, because you're on the second term is gonna be one. That's one which is one. So now we differentiate, It's your directive together forthe derivative. Why again? Combination of product rule in general half minus a sign off acts plus y time swan plus y prime coop minus sign of X plus y. So since we can't side effects, that's why we already know that asshole turn gonna be multiplied by zero and disappear. But for the sake of completeness, that's computed it is too second Jewish for why? Damn swan Los y prime a minus Sign X plus y time Swan plus y prime damp. Second judge for why? Plus sign off exports. Why Time surgery? Usual. Foy. So we substitute zero everywhere here. So when did you get so this term is gonna be minus one. His German's gonna disappear. It's multiplied by zero. Same. Is this true? And here we have plus one. So the four story which of our fly and zero equals to zero. So we just need one more term. Let's try to compute Why fizzed original work? So it becomes sign of X plus y I guess. One plus y prime over four minus sign off X plus y time three Gas one plus Why Prime squirt Second Jewish For why, minus a sign X plus y times one plus y prime squared. That was too wise. A two second time Second George for white minus two. Sign off X plus y expression becomes quite lengthy, But again most rose will disappear. For example, this one with just computed for state of completeness. Surgical. Why does one plus what a prime plus second village for y squared minus minus A sign of X plus y, I guess. One plus y prime squirt, You have a second George for why minus sign off X plus y times. Big Square bracket here. Same term. Why Double Prime Squared last? Why Triple Prime? Just one loss. White Pride minus Sign off, X plus y second George For Why, Plus a sign, X plus y four's The Ridge Malfoy. It's No, it's substitute zero everywhere, and we're careful was getting older terms correctly. We get zero minus three terms. Sign of zero. Just one my desk sign of zero times two minus x sign of zero. There's one and the rest of the terms zero. So we, in fact, half minus three minus two, minus one minus six. So now let's open a new page and get it old guesser. So why is approximately X squared or two plus X cube over six minor six times Export five or five factorial, which is two times three times four times five or simply excluded, or two plus X cube or six minus export five or 20. And that is the final answer

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