Dania Robbins

2022-04-23

Solution of simultaneous ODE :

${x}^{\prime}=({x}^{2}+{y}^{2})y$ and ${y}^{\prime}=-({x}^{2}+{y}^{2})x$

tswe0uk

Beginner2022-04-24Added 19 answers

Step 1

You can assemble the two equations into a complex-scalar equation as

$\dot{z}=-i{\left|z\right|}^{2}z,\text{}z=x+iy.$

As observed, the radius has then the equation

$r\dot{r}=\frac{12}{\frac{d}{dt}}{\left|z\right|}^{2}=Re\left(\stackrel{\u2015}{z}\dot{z}\right)=Re(-i){\left|z\right|}^{4}=0,$

giving $r=c$ constant. Then the original equation has a simple linear form

$\dot{z}=\lambda z,\text{}\lambda =-i{c}^{2}$

$\Rightarrow z\left(t\right)={z}_{0}{e}^{-i{c}^{2}t},\text{}\left|{z}_{0}\right|=c\text{}\text{or}\text{}{z}_{0}=c{e}^{i\varphi}$

$\Rightarrow x\left(t\right)+iy\left(t\right)=c,(\mathrm{cos}({c}^{2}t-\varphi )-i\mathrm{sin}({c}^{2}t-\varphi ))$

What is the derivative of the work function?

How to use implicit differentiation to find $\frac{dy}{dx}$ given $3{x}^{2}+3{y}^{2}=2$?

How to differentiate $y=\mathrm{log}{x}^{2}$?

The solution of a differential equation y′′+3y′+2y=0 is of the form

A) ${c}_{1}{e}^{x}+{c}_{2}{e}^{2x}$

B) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{3x}$

C) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{-2x}$

D) ${c}_{1}{e}^{-2x}+{c}_{2}{2}^{-x}$How to find instantaneous velocity from a position vs. time graph?

How to implicitly differentiate $\sqrt{xy}=x-2y$?

What is 2xy differentiated implicitly?

How to find the sum of the infinite geometric series given $1+\frac{2}{3}+\frac{4}{9}+...$?

Look at this series: 1.5, 2.3, 3.1, 3.9, ... What number should come next?

A. 4.2

B. 4.4

C. 4.7

D. 5.1What is the derivative of $\frac{x+1}{y}$?

How to find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?

How to find the volume of a cone using an integral?

What is the surface area of the solid created by revolving $f\left(x\right)={e}^{2-x},x\in [1,2]$ around the x axis?

How to differentiate ${x}^{\frac{2}{3}}+{y}^{\frac{2}{3}}=4$?

The differential coefficient of $\mathrm{sec}\left({\mathrm{tan}}^{-1}\left(x\right)\right)$.