Daniaal Sanchez
2020-12-17
Answered

Solve differential equation ${L}^{\prime}(x)=k(x+b)(L-a)$

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asked 2022-01-20

Solve and find the soltion to the first order differential equation:

$x\frac{dy}{dx}+y={e}^{x},\text{}\text{}x0$

asked 2022-07-09

Find differential equation ${y}^{\mathrm{\prime}}=f(t,y)$ satisfied by $y(t)=4\phantom{\rule{thinmathspace}{0ex}}{e}^{2t}+3$

Solution:

Compute derivative of y,

${y}^{\mathrm{\prime}}=8\phantom{\rule{thinmathspace}{0ex}}{e}^{2t}$

Write right hand side above, in terms of the original function y, that is,

$y-3=4\phantom{\rule{thinmathspace}{0ex}}{e}^{2t}$

Get a differential equation satisfied by y, namely

${y}^{\mathrm{\prime}}=2y-6$

So my issue with that last answer. How is this a solution? Does it mean that if you somehow take an integral of $2y-6$ you should end up with the original $y(t)=4\phantom{\rule{thinmathspace}{0ex}}{e}^{2t}+3$???

It seems that there two different derivatives of y(t)

one is:

${y}^{\mathrm{\prime}}=8\phantom{\rule{thinmathspace}{0ex}}{e}^{2t}$

the other is:

${y}^{\mathrm{\prime}}=2y-6$

and I don't get it, can someone explain?

Also a bit offtopic, but the way y'=f(t,y) is written kinda bugs me.

Shouldn't it be written like y'=f(t,y(t)) to show that the function f contains t as an independent variable and the function y(t) which contains variable t as an input to itself (dependent variable t) ??? That's kinda an essential information, so surprised it's omitted in the writings.

Solution:

Compute derivative of y,

${y}^{\mathrm{\prime}}=8\phantom{\rule{thinmathspace}{0ex}}{e}^{2t}$

Write right hand side above, in terms of the original function y, that is,

$y-3=4\phantom{\rule{thinmathspace}{0ex}}{e}^{2t}$

Get a differential equation satisfied by y, namely

${y}^{\mathrm{\prime}}=2y-6$

So my issue with that last answer. How is this a solution? Does it mean that if you somehow take an integral of $2y-6$ you should end up with the original $y(t)=4\phantom{\rule{thinmathspace}{0ex}}{e}^{2t}+3$???

It seems that there two different derivatives of y(t)

one is:

${y}^{\mathrm{\prime}}=8\phantom{\rule{thinmathspace}{0ex}}{e}^{2t}$

the other is:

${y}^{\mathrm{\prime}}=2y-6$

and I don't get it, can someone explain?

Also a bit offtopic, but the way y'=f(t,y) is written kinda bugs me.

Shouldn't it be written like y'=f(t,y(t)) to show that the function f contains t as an independent variable and the function y(t) which contains variable t as an input to itself (dependent variable t) ??? That's kinda an essential information, so surprised it's omitted in the writings.

asked 2022-09-29

What is the solution to the Differential Equation $\frac{4}{{y}^{3}}\frac{dy}{dx}=\frac{1}{x}$?

asked 2022-07-01

The differential equation that describes my system is given as

${y}^{(n)}+{a}_{n-1}{y}^{(n-1)}+\cdots +{a}_{1}\dot{y}+{a}_{0}y={b}_{n-1}{u}^{(n-1)}+\cdots +{b}_{1}\dot{u}+{b}_{0}u+g(y(t),u(t))$

I want to express the above differential equation into a system of linear differential equations of the form

$\dot{x}=Ax+Bu+{B}_{p}g$

$y=Cx$

The matrices are given as follows: However, I am not able to prove, how to get them

$A=\left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ -{a}_{0}& -{a}_{1}& -{a}_{2}& \cdots & -{a}_{n-1}\end{array}\right]$

$C=\left[\begin{array}{ccccc}1& {b}_{1}/{b}_{0}& {b}_{2}/{b}_{0}& \cdots & {b}_{n-1}/{b}_{0}\end{array}\right]$

How do I get the above matrices from the differential equation form as shown above?

${y}^{(n)}+{a}_{n-1}{y}^{(n-1)}+\cdots +{a}_{1}\dot{y}+{a}_{0}y={b}_{n-1}{u}^{(n-1)}+\cdots +{b}_{1}\dot{u}+{b}_{0}u+g(y(t),u(t))$

I want to express the above differential equation into a system of linear differential equations of the form

$\dot{x}=Ax+Bu+{B}_{p}g$

$y=Cx$

The matrices are given as follows: However, I am not able to prove, how to get them

$A=\left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ -{a}_{0}& -{a}_{1}& -{a}_{2}& \cdots & -{a}_{n-1}\end{array}\right]$

$C=\left[\begin{array}{ccccc}1& {b}_{1}/{b}_{0}& {b}_{2}/{b}_{0}& \cdots & {b}_{n-1}/{b}_{0}\end{array}\right]$

How do I get the above matrices from the differential equation form as shown above?

asked 2022-06-01

Is a first order differential equation categorized by $f({y}^{\prime},y,x)=0$ or ${y}^{\prime}=f(y,x)$?

In the case of the second, why $\mathrm{sin}{y}^{\prime}+3y+x+5=0$ isnt a first order differential equation?

In the case of the second, why $\mathrm{sin}{y}^{\prime}+3y+x+5=0$ isnt a first order differential equation?

asked 2022-06-23

I want to simulate the behaviour of a 2-DOF robotic manipulator, which is described by the following model:

$M(q)\ddot{q}=-C(q,\dot{q})\dot{q}-G(q)+\tau \text{}\text{}\text{}\text{}\text{}(1)$

Considering the fact that the 2x2 mass matrix M is positive definite, I could use the inverse matrix and break down the problem into the 4 first order ordinary differential equations and simulate it:

${x}_{1}={q}_{1}\Rightarrow \dot{{x}_{1}}={x}_{2}$

${x}_{2}=\dot{{q}_{1}}\Rightarrow \dot{{x}_{2}}=-{M}^{-1}(1,:)\cdot C\cdot {\left[\begin{array}{cc}{x}_{2}& {x}_{4}\end{array}\right]}^{T}-{M}^{-1}(1,:)\cdot G+{M}^{-1}(1,:)\cdot \tau $${x}_{3}={q}_{2}\Rightarrow \dot{{x}_{3}}={x}_{4}$

${x}_{4}=\dot{{q}_{2}}\Rightarrow \dot{{x}_{4}}=-{M}^{-1}(2,:)\cdot C\cdot {\left[\begin{array}{cc}{x}_{2}& {x}_{4}\end{array}\right]}^{T}-{M}^{-1}(2,:)\cdot G+{M}^{-1}(2,:)\cdot \tau $

Suppose I would like to use a solver that takes as an argument the mass matrix (a MATLAB ODE solver in particular) and don't use its inverse because this will also simplify the computation of the jacobian (I intend to simulate a 7-DOF robotic manipulator after that so providing the mass matrix would be great). How can I write the initial equation (1) as a series of first order ordinary differential equations and be able to simulate its response by using some software solvers ?

$M(q)\ddot{q}=-C(q,\dot{q})\dot{q}-G(q)+\tau \text{}\text{}\text{}\text{}\text{}(1)$

Considering the fact that the 2x2 mass matrix M is positive definite, I could use the inverse matrix and break down the problem into the 4 first order ordinary differential equations and simulate it:

${x}_{1}={q}_{1}\Rightarrow \dot{{x}_{1}}={x}_{2}$

${x}_{2}=\dot{{q}_{1}}\Rightarrow \dot{{x}_{2}}=-{M}^{-1}(1,:)\cdot C\cdot {\left[\begin{array}{cc}{x}_{2}& {x}_{4}\end{array}\right]}^{T}-{M}^{-1}(1,:)\cdot G+{M}^{-1}(1,:)\cdot \tau $${x}_{3}={q}_{2}\Rightarrow \dot{{x}_{3}}={x}_{4}$

${x}_{4}=\dot{{q}_{2}}\Rightarrow \dot{{x}_{4}}=-{M}^{-1}(2,:)\cdot C\cdot {\left[\begin{array}{cc}{x}_{2}& {x}_{4}\end{array}\right]}^{T}-{M}^{-1}(2,:)\cdot G+{M}^{-1}(2,:)\cdot \tau $

Suppose I would like to use a solver that takes as an argument the mass matrix (a MATLAB ODE solver in particular) and don't use its inverse because this will also simplify the computation of the jacobian (I intend to simulate a 7-DOF robotic manipulator after that so providing the mass matrix would be great). How can I write the initial equation (1) as a series of first order ordinary differential equations and be able to simulate its response by using some software solvers ?

asked 2021-09-24

Verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit.

a)

b)

c)