For these six numbers, e3, 3e, eπ, πe, 3π, π3, how to arrange them in the increasing order? This problem is taken from the today test: National Higher Education Entrance Examination.

If you simplify by factoring out $e^ {2t}$ and cancelling, that would give you $$g' - 2g = 3e^ {2t},$$ instead of $g'-2g = 3e^ {4t}$, which is what you had. So it looks like you made a simplification …

Sep 9, 2024 · Solving $\int_0^ {\infty}x^3e^ {-x^2}dx$ [duplicate] Ask Question Asked 1 year, 3 months ago Modified 1 year, 3 months ago

Nov 7, 2021 · Find the general solution to $xy' = 2y + x^3e^x$ Ask Question Asked 4 years, 1 month ago Modified 4 years, 1 month ago

Nov 17, 2014 · $$5+55+555+5555+\cdots+\overbrace {55\dots5}^ {n\text { fives}}$$ $$=\frac59 (9+99+999+9999+\cdots+\overbrace {99\dots9}^ {n\text { nines}})$$ $$=\frac59 (10^1-1+10^2 ...

Aug 14, 2015 · A nice and quick way to visualize integration by parts (it could be a time-saver!): $$\matrix {&\text {differentiate}&&& &\text {integrate}&\\ &x^3&&&&e^x ...

Apr 8, 2016 · Taking the Laplace Transform to both sides of the ODE we get \begin {align*} s^2Y (s)-s (1)- (-1)-4 [sY (s)-1]&=\frac 6 {s-3}-\frac 3 {s+1}\\ [4pt] (s^2-4s)Y (s ...

Feb 3, 2016 · Evaluation of $\int_ {0}^ {\infty}t^3e^ {-3t}dt$ Ask Question Asked 9 years, 10 months ago Modified 9 years, 10 months ago

However, what you’ve written is incorrect, starting with the missing $dz$ in the integral: $\int z^3e^z\,\underline {dz}$. For your integration by parts $u=z^3$, not $z^3\,dz$, and …

$$A (2t\cdot 3e^ {3t}+t^2\cdot 9e^ {3t}+2e^ {3t}+6te^ {3t}+9 (At^2e^ {3t}) = t^2e^ {3t}$$ but how should I group the terms in the left to equate to the ones in the right?