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# Use power series to solve the differential equation.$y' = xy$

## $y=c_{0} \cdot e^{x^{2} / 2}$

#### Topics

Second-Order Differential Equations

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### Video Transcript

today we're going to solve a problem. But yes, if there is a solution off the form by equals sigma and equal to zero to infinity seeing exist to it Different ships this one with respect expired Isaac was sigma and nickel toe one to infinity in feet and access toe end minus one differential equation. We need to solve this. Wild ice equals X y. So replace violence and live with the submission We will get like state MMA unequal toe one to infinity in C and access during minus one, which is equal toe X into sigma and negative 02 in 20 Seeing eggs. Restaurant sigma A negative one 2030 and seeing X rays to end minus one equals sigma in equal to zero to infinity. See and x rays toe and plus one C one plus sigma unequal toe to infinity and see And it says two in minus Monica's sigma in equal to zero to infinity. Seeing access to and plus one okay to Stephen Plus Sigma and equal to zero to infinity and plus two seeing plus two excess doing plus one equals Sigma Clinical 20 to infinity. Seeing existing plus one say what I equal toe a Toby A 45 equals Sigma I equals a mine escape B minus k it forth a plus k see one plus sigma any go to zero to infinity. Thank plus two seat and plus two. It's restaurant plus one minus. So come on, I need go to zero to infinity, seeing existing plus one equal to zero. See when plus Sigma and equal to zero to infinity end Plus to see and just to express doing plus one my not seeing expressed in plus one equal to zero stealing plus Sigma unequal to zero to infinity and plus two c n plus two minus. Seeing picture is doing less. One equals zero for the sun to the zero we need to see one equals zero for en plus two c n plus two minus ian equals zero see and plus two equals C and developed by and plus two substitute chemicals. One C three was see when they were hit by one plus two, which is zero. They were by three, which is zero. See flight equals C really worried by three plus two, which is zero by five, which is zero. Similarly, C one for all are all terms will be zero our terms. Uh huh. Zero Then he to c two because they not by two c four equals C two by four the physical to see not by do in the food C six equals C four by six, which is C not by two into four in those six. So si tu n equals c zero divided by two into four into six Hindu to it. See doing acres one by two or to the powering dot CEO divided by one in two in three like in the end c two. And it was one way to hold the parent industry no different by in factorial so right equals sigma. Any go to the European affinity seeing express train which is equal to signal an equal to zero to infinity See you to win Existed to and yes Sigma and equal to zero to infinity See off to impress one excess to two and plus one It was difficult to sigma any go to zero to infinity one going toe hold the parent See not diverted by in factorial access to to when plus sigma any go to 0 18 50 zero and do express to do and plus one So which is equal toe. See not into Sigma n equals zero to infinity one day toe. Hold their parents in do. It's a square hole to the border and developed by in factorial, which is a chemical toe. C zero Indo sigma en equal to zero to infinity one by to hold to the parents, Take the square hold up or and they went by in Victoria, which is equal to C zero sigma Any good zero to infinity X square by two holder Parent doing great Infected it The physical to see zero in the areas two X squared by two That standoff Listen. Thank you.

University of New Mexico

#### Topics

Second-Order Differential Equations

Lectures

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