(b) Using Green's int c x^ 2 ydx + x ^ 2 * dxy theorem, evaluate where C is the triangle

joining the points (0, 0), (1, 0) and (0, 1).

Priya
2022-07-31

(b) Using Green's int c x^ 2 ydx + x ^ 2 * dxy theorem, evaluate where C is the triangle

joining the points (0, 0), (1, 0) and (0, 1).

You can still ask an expert for help

asked 2021-01-02

Find

asked 2022-04-23

Given: $x{y}^{{}^{\u2033}}+(2+2x){y}^{{}^{\prime}}+2y=0$ has a solution $y}_{1}\left(x\right)={x}^{-1$ , find the general solution of the differential equation $x{y}^{{}^{\u2033}}+(2+2x){y}^{{}^{\prime}}+2y=8{e}^{2x}$

asked 2022-03-21

Let $f\in {C}_{o}^{\mathrm{\infty}}\left({\mathbb{R}}^{n}\right)$ . Propose formulas of the form $u\left(x\right)=displaysty\le {\int}_{\{}^{\{}\frac{\hat{f}\left(\delta \right){e}^{ix\delta}}{p\left(\delta \right)}d\delta$

to solve:

i)$\sum _{j=1}^{n}\frac{{\partial}^{4}u\left(x\right)}{\partial {x}_{j}^{4}}+u\left(x\right)=f\left(x\right)$

andii)$\sum _{k,l}^{\{}\frac{{\partial}^{4}u\left(x\right)}{\partial {x}_{k}^{2}\partial {x}_{l}^{2}}-2\cdot displaysty\le \sum _{k=1}^{n}\frac{{\partial}^{2}u\left(x\right)}{\partial {x}_{k}^{2}}+u\left(x\right)=f\left(x\right)$

to solve:

i)

andii)

asked 2022-03-20

Solve ${x}^{3}\left\{y\right\}{}^{\u2034}+x{\left\{y\right\}}^{\prime}-y=x\mathrm{ln}\left(x\right)$

using shift $x={e}^{z}$ and differential operator $Dz=\frac{d}{dz}$.

What does $Dz=\frac{d}{dz}$ mean?

${\left({e}^{z}\right)}^{3}\left\{y\right\}{}^{\text{'}\text{'}\text{'}}+{e}^{z}{\left\{y\right\}}^{\text{'}}-y={e}^{z}\mathrm{ln}\left({e}^{z}\right)$

$\left({e}^{3z}\right){\left\{y\right\}}^{\text{'}\text{'}\text{'}}+{e}^{z}{\left\{y\right\}}^{\text{'}}-y={e}^{z}\left\{z\right\}$

$y={z}^{r}$

${e}^{3}r({r}^{2}-r-2){z}^{r-3}+{e}^{z}r{z}^{r-1}-{z}^{r}=0$

Continue ?

asked 2022-04-21

Variation of parameters method for differential equations.

Change the variable$x={e}^{t}$ and then find the general solution for the following differential equation $2{x}^{2}y{}^{\u2033}-6x{y}^{\prime}+8y=2x+2{x}^{2}\mathrm{ln}x$

It seems a little suspicious that i can factor out 2 and x first.${x}^{2}y{}^{\u2033}-3x{y}^{\prime}+4y=x(\mathrm{ln}\left(x\right)+1)$ and if we factor out $x}^{2$ now we end up with $y{}^{\u2033}-\frac{3}{x}{y}^{\prime}+\frac{4}{{x}^{2}}y=\frac{x\mathrm{ln}x+1}{x}$

(1) Therefore substituting$x={e}^{t}$ (1) now becomes $y{}^{\u2033}-\frac{3}{{e}^{t}}{y}^{\prime}+\frac{4}{{e}^{2t}}y=\frac{{e}^{\mathtt{+}}1}{{e}^{t}}$ . We need constant coefficients in order to use the variation of parameters method am i right?

Change the variable

It seems a little suspicious that i can factor out 2 and x first.

(1) Therefore substituting

asked 2022-03-19

Can this ODE system be solved?

$x}^{\prime}\left(t\right)=\mathrm{sin}(x\left(t\right)(\frac{y\left(t\right)}{2}+1);{y}^{\prime}\left(t\right)=\frac{-\mathrm{cos}\left(x\left(t\right)\right)\mathrm{cos}\left(y\left(t\right)\right)}{2$

Is there a method to solve the following ODE system?

with initial conditions $x\left(0\right)={x}_{0}\text{}\text{and}\text{}y\left(0\right)={y}_{0}$.

asked 2022-04-07

Finding a constant such that the following integral inequality holds:

Constant: $c>0$ such that for all $C}^{1$ function in $(0,1)$

$cu{\left(0\right)}^{2}\le {\int}_{0}^{1}{u}^{\prime 2}+{u}^{2}dt$