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 ...

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

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Nov 19, 2019 · 2 For the particular solution, $3e^t$ method of undetermined coefficients can be used. Try $At^2e^t$.

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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?

Sep 9, 2024 · You used a substitution, and then integration by parts once. But what if you start out with integration by parts? $$ \begin {align} \int x^3e^ {-x^2}\,dx &=\int -\frac12x^2\left (-2xe^ { …

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 ...

Sep 7, 2016 · Given a random variable $X\\sim\\mathcal N(0,\\sigma^2)$, how can we prove that $E[X^4]=3\\sigma^4$? I am having trouble even starting with the proof.