How can I start to solve this differential equation?

${y}^{\prime}=\frac{y}{2y\mathrm{ln}(y)+y-x}$

${y}^{\prime}=\frac{y}{2y\mathrm{ln}(y)+y-x}$

ngekhonofn1lv
2022-06-01
Answered

How can I start to solve this differential equation?

${y}^{\prime}=\frac{y}{2y\mathrm{ln}(y)+y-x}$

${y}^{\prime}=\frac{y}{2y\mathrm{ln}(y)+y-x}$

You can still ask an expert for help

Schezzix8la5

Answered 2022-06-02
Author has **2** answers

Hint

You could notice that the equation simplifies if you consider $x$ as a function of $y$. So

${y}^{\prime}=\frac{y}{2y\mathrm{ln}(y)+y-x}$

becomes

${x}^{\prime}+\frac{x}{y}=1+2\mathrm{log}(y)$

I am sure that you can take from here.

You could notice that the equation simplifies if you consider $x$ as a function of $y$. So

${y}^{\prime}=\frac{y}{2y\mathrm{ln}(y)+y-x}$

becomes

${x}^{\prime}+\frac{x}{y}=1+2\mathrm{log}(y)$

I am sure that you can take from here.

asked 2022-09-17

How do you find the differential dy of the function $y=x\sqrt{1-{x}^{2}}$?

asked 2022-05-15

I am given this:

$(2x+1)\frac{dy}{dx}+y=0$

I tried this:

$\frac{1}{(2x+1)}dx=\frac{-1}{y}dy$

Then integrated the above sum and got this:

$\frac{ln(2x+1)}{2}=-ln(y)$

The answer is: ${y}^{2}(2x+1)=C$.

I tried solving it by placing the like terms together and integrating them. However, my answer is wrong from the answer given. Could you point out my mistake? Or am i evaluating the entire thing incorrectly?

All suggestions and help are appreciated!

$(2x+1)\frac{dy}{dx}+y=0$

I tried this:

$\frac{1}{(2x+1)}dx=\frac{-1}{y}dy$

Then integrated the above sum and got this:

$\frac{ln(2x+1)}{2}=-ln(y)$

The answer is: ${y}^{2}(2x+1)=C$.

I tried solving it by placing the like terms together and integrating them. However, my answer is wrong from the answer given. Could you point out my mistake? Or am i evaluating the entire thing incorrectly?

All suggestions and help are appreciated!

asked 2020-12-29

Solve differential equation$x{y}^{\prime}+[(2x+1)/(x+1)]y=x-1$

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What is a solution to the differential equation $xy\prime =y$?

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Solve for G.S./P.S. for the following differential equations using the solutions by a change in variables as suggested by the equations.

$({t}^{2}{e}^{t}-4{x}^{2})dt+8txdx=0$

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 ?

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What is a solution to the differential equation $y\prime =\frac{1}{2}\mathrm{sin}\left(2x\right)$?