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can-t-Solve-Differantial-Equation-Diff-Equa-dy-t-dt-2-4y-t-8t-2-32t-28-Sadly-it-s-impossible-to-obtain-an-exact-closed-form-expression-of-the-Solution-of-Diff-Equa-But-if-the-Runge-Kut




Question Number 205833 by MathedUp last updated on 31/Mar/24
can′t Solve Differantial Equation  Diff Equa : (((dy(t))/dt))^2 +4y(t)=8t^2 −32t+28....  Sadly it′s impossible to obtain an exact  closed−form expression of the Solution of Diff Equa  But if the Runge−Kutta method is used.  the value of the function at any one point can be estimated
$$\mathrm{can}'\mathrm{t}\:\mathrm{Solve}\:\mathrm{Differantial}\:\mathrm{Equation} \\ $$$$\mathrm{Diff}\:\mathrm{Equa}\::\:\left(\frac{\mathrm{d}{y}\left({t}\right)}{\mathrm{d}{t}}\right)^{\mathrm{2}} +\mathrm{4}{y}\left({t}\right)=\mathrm{8}{t}^{\mathrm{2}} −\mathrm{32}{t}+\mathrm{28}…. \\ $$$$\mathrm{Sadly}\:\mathrm{it}'\mathrm{s}\:\mathrm{impossible}\:\mathrm{to}\:\mathrm{obtain}\:\mathrm{an}\:\mathrm{exact} \\ $$$$\mathrm{closed}−\mathrm{form}\:\mathrm{expression}\:\mathrm{of}\:\mathrm{the}\:\mathrm{Solution}\:\mathrm{of}\:\mathrm{Diff}\:\mathrm{Equa} \\ $$$$\mathrm{But}\:\mathrm{if}\:\mathrm{the}\:\mathrm{Runge}−\mathrm{Kutta}\:\mathrm{method}\:\mathrm{is}\:\mathrm{used}. \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function}\:\mathrm{at}\:\mathrm{any}\:\mathrm{one}\:\mathrm{point}\:\mathrm{can}\:\mathrm{be}\:\mathrm{estimated} \\ $$
Commented by mr W last updated on 31/Mar/24
y(t)=t^2 −4t+3  or  y(t)=−2t^2 +8t−9
$${y}\left({t}\right)={t}^{\mathrm{2}} −\mathrm{4}{t}+\mathrm{3} \\ $$$${or} \\ $$$${y}\left({t}\right)=−\mathrm{2}{t}^{\mathrm{2}} +\mathrm{8}{t}−\mathrm{9} \\ $$
Commented by MathedUp last updated on 31/Mar/24
r u assumed y(t) as at^2 +bt+c ????  (2at+b)^2 +4at^2 +4bt+4c=8t^2 −32t+28
$$\mathrm{r}\:\mathrm{u}\:\mathrm{assumed}\:{y}\left({t}\right)\:\mathrm{as}\:{at}^{\mathrm{2}} +{bt}+{c}\:???? \\ $$$$\left(\mathrm{2}{at}+{b}\right)^{\mathrm{2}} +\mathrm{4}{at}^{\mathrm{2}} +\mathrm{4}{bt}+\mathrm{4}{c}=\mathrm{8}{t}^{\mathrm{2}} −\mathrm{32}{t}+\mathrm{28} \\ $$
Commented by mr W last updated on 31/Mar/24
why not?
$${why}\:{not}? \\ $$
Commented by MathedUp last updated on 31/Mar/24
Oh.... yeah you were right...  i thought he was looking for a more general solution  :(
$$\mathrm{Oh}….\:\mathrm{yeah}\:\mathrm{you}\:\mathrm{were}\:\mathrm{right}… \\ $$$$\mathrm{i}\:\mathrm{thought}\:\mathrm{he}\:\mathrm{was}\:\mathrm{looking}\:\mathrm{for}\:\mathrm{a}\:\mathrm{more}\:\mathrm{general}\:\mathrm{solution} \\ $$$$:\left(\right. \\ $$
Commented by mr W last updated on 31/Mar/24
there might be other solutions.
$${there}\:{might}\:{be}\:{other}\:{solutions}. \\ $$

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