Molecca89g

2022-04-29

I have encountered a question that goes like

$\frac{dy}{dx}={y}^{3}+x\mathrm{cos}\left(x\right),\text{}y\left(0\right)=0.$

Find the approximate solution which is$o\left({x}^{5}\right)$

Find the approximate solution which is

kayuukor9c

Beginner2022-04-30Added 13 answers

You insert a (truncated) power series up to the 5th order. To determine these coefficients, you need to equate coefficients up to the 4th power in

${a}_{1}+2{a}_{2}x+3{a}_{3}{x}^{2}+4{a}_{4}{x}^{3}+5{a}_{5}{x}^{4}+o\left({x}^{4}\right)$

$={({a}_{0}+{a}_{1}x+\dots +{a}_{4}{x}^{4}+o\left({x}^{4}\right))}^{3}+x-\frac{{x}^{3}}{2}+o\left({x}^{4}\right).$

As everything goes well in this power series expansion, one could also replace$o\left({x}^{4}\right)$ with the sharper $O\left({x}^{5}\right)$ .

As everything goes well in this power series expansion, one could also replace

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