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Author: Tinku Tara

Solve-2-w-t-2-c-2-2-w-x-2-w-0-t-f-t-lim-x-w-x-t-0-Boundary-Condition-w-x-0-0-w-t-x-0-0-Initial-Condition-f-t-sin-t-t-0-2pi-0-otherwise-

Question Number 218539 by SdC355 last updated on 11/Apr/25 $${S}\mathrm{olve} \\ $$$$\frac{\partial^{\mathrm{2}} {w}}{\partial{t}^{\mathrm{2}} }={c}^{\mathrm{2}} \frac{\partial^{\mathrm{2}} {w}}{\partial{x}^{\mathrm{2}} } \\ $$$${w}\left(\mathrm{0},{t}\right)={f}\left({t}\right)\:,\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{w}\left({x},{t}\right)=\mathrm{0}\:\left(\mathrm{Boundary}\:\mathrm{Condition}\right) \\ $$$${w}\left({x},\mathrm{0}\right)=\mathrm{0}\:,\:{w}_{{t}} \left({x},\mathrm{0}\right)=\mathrm{0}\:\left(\mathrm{Initial}\:\mathrm{Condition}\right) \\ $$$${f}\left({t}\right)\begin{cases}{\mathrm{sin}\left({t}\right)\:,\:{t}\in\left[\mathrm{0},\mathrm{2}\pi\right)}\\{\mathrm{0}\:,\:\mathrm{otherwise}}\end{cases}…

I-Find-Fun-integral-problem-x-dx-1-note-I-already-know-that-integral-Solution-Try-integral-problem-Product-Integral-Integral-

Question Number 218534 by SdC355 last updated on 11/Apr/25 $$\mathrm{I}\:\mathrm{Find}\:\mathrm{Fun}\:\mathrm{integral}\:\mathrm{problem}! \\ $$$$\int\:\:{x}^{\mathrm{d}{x}} −\mathrm{1}\:=??\: \\ $$$$\left.\mathrm{note}\right)\:\:\mathrm{I}\:\mathrm{already}\:\mathrm{know}\:\mathrm{that}\:\mathrm{integral}\:\mathrm{Solution} \\ $$$$\:\mathrm{Try}\:\mathrm{integral}\:\mathrm{problem}! \\ $$$$\left(#\:\mathrm{Product}\:\mathrm{Integral}\:,\:#\mathrm{Integral}\right) \\ $$ Terms of Service Privacy…

Question-218494

Question Number 218494 by Spillover last updated on 10/Apr/25 Answered by vnm last updated on 10/Apr/25 $$\frac{{R}}{\:\sqrt{{R}^{\mathrm{2}} +\left(\mathrm{2}{R}\right)^{\mathrm{2}} }}=\frac{{r}}{\:\sqrt{{R}^{\mathrm{2}} +\left(\mathrm{2}{R}\right)^{\mathrm{2}} }−\left({R}+{r}\right)} \\ $$$$\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}}}=\frac{{r}}{\:{R}\sqrt{\mathrm{5}}−{R}−{r}}=\frac{{r}/{R}}{\:\sqrt{\mathrm{5}}−\mathrm{1}−{r}/{R}} \\ $$$$\sqrt{\mathrm{5}}=\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{{r}/{R}}−\mathrm{1}…

solve-for-real-x-3-x-5-5x-4-10x-3-10x-2-5x-x-2-2x-1-x-4-4x-2-4-3-

Question Number 218482 by Nicholas666 last updated on 10/Apr/25 $$ \\ $$$$\:\:\:\:{solve}\:{for}\:{real}\:\boldsymbol{{x}}; \\ $$$$ \\ $$$$\underset{\:\:\:\mathrm{3}\sqrt{\frac{\boldsymbol{{x}}^{\mathrm{5}} −\mathrm{5}\boldsymbol{{x}}^{\mathrm{4}} +\mathrm{10}\boldsymbol{{x}}^{\mathrm{3}} −\mathrm{10}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\boldsymbol{{x}}}{\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{x}}+\mathrm{1}}+\sqrt{\boldsymbol{{x}}^{\mathrm{4}} +\mathrm{4}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{4}}}=\sqrt{\mathrm{3}\:}\:\:\:} {\:} \\…