Recent content by Sudharaka

Hi, This is a separable differential equation; and more information about separable differential equations are explained here, Differential Equations - Separable Equations To solve this you can write it as, $$y\frac{dy}{dx}=x^2$$ $$\Rightarrow \int y dy = \int x^2 dx$$ Hope you can...

Hi Another, :) Interesting question, thanks. Notice that, $${P}_{n}(x) = \frac{1}{2^nn!}\frac{d^n}{dx^n}(x^2-1)^n=\frac{(2x)n}{2^n n!}\frac{d^{n-1}}{dx^{n-1}}(x^2 - 1)^{n-1}$$ and so on and generally for a differentiation of $r$ times we get, $${P}_{n}(x) = \frac{(2x)^{r}}{2^n...

Yes that is correct. Well done. :)

Hi, Notice that, $\frac{d}{dt}\left(e^{-t}y\right)=e^{-t}y'-ye^{-t}$. I hope you can continue with this hint. :)

No worries. :) Also I note that the provided answer is wrong; it should be, https://www.wolframalpha.com/input/?i=ty%27-y%3D(t%5E2)%2Fe

Hi karush, :) I think you have done a minor mistake; observe that; $$t\left(\frac{y}{t}\right)'=y'-\frac{y}{t}$$. Hope you can do it from here :)

Hi RTCNTC, 1) There's no hard rule on how many values you should use. The more values you use the smoother/accurate the graph would be. 2) Again this depends on which domain you want to draw the graph in. I would choose $x$ values from -10 to +10 with increments of 1 (total of 20 values)...

Yes, since the volume of each square is given by $2x\,dx$ the volume is given by the integral $\int_0^4 2x\,dx$.

Hi karush, Yes. Since the length of the diagonal of the square is $2\sqrt{x}$ if we take $y$ as the length of its side, by the Pythagorean theorem we have $y^2 + y^2=\left(2\sqrt{x}\right)^2\Rightarrow y^2=2x$. Therefore the area of the square is $2x$.

Hi tmt, :) Here's a hint. Divide both the numerator and the denominator of the fraction inside the denominator by $n$ so that you get, $${a}_{n} = \ln \left(\frac{12 + \frac{2}{n}}{-\frac{9}{n} + 4}\right)$$ Try to continue from here. :)

Hi shameih, It is true that $\begin{bmatrix}-12\\-8\\20\end{bmatrix}$ is a non-zero vector that is in the column space of $M$. I see nothing wrong with your answer. Is it possible that the way you inserted it into the computer might be incorrect?

Hi shamieh, The two columns of the matrix are $$\begin{pmatrix}1\\0\end{pmatrix}$$ and $$\begin{pmatrix}0\\0\end{pmatrix}$$. To check the linear independence of these two vectors take $$\alpha$$ and $$\beta$$ such that...

Hi uNmiN, :) $$\lim_{n\rightarrow\infty}\frac{\sin(nx)}{\sin x}=\frac{1}{\sin x}\lim_{n\rightarrow\infty}\sin(nx)$$ For a fixed value $x$, $\sin(nx)$ alternates sign as $n$ changes. Therefore the limit of $\sin(nx)$ as $n$ goes to infinity does not exist. Thus the limit...

Hi NotaMathPerson, There's a method to find the cube root of a number, similar to long division mentioned at the following webpage. It might be the one you are looking for. https://xlinux.nist.gov/dads//HTML/cubeRoot.html