Identify the following differential equation and hence solve it $y\prime =-\frac{4}{{x}^{2}}-\frac{y}{x}+{y}^{2}$?

pramrok62
2022-09-29
Answered

Identify the following differential equation and hence solve it $y\prime =-\frac{4}{{x}^{2}}-\frac{y}{x}+{y}^{2}$?

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Nolan Tyler

Answered 2022-09-30
Author has **9** answers

We have:

$y\prime =-\frac{4}{{x}^{2}}-\frac{y}{x}+{y}^{2}$.... [A]

This is a non-linear first order Differential Equation. We can attempt a substitution:

$v=xy\iff y=\frac{v}{x}$

And differentiating using the product rule we get:

$\frac{dv}{dx}=x\frac{dy}{dx}+y\Rightarrow \frac{dy}{dx}=\frac{\frac{dv}{dx}-y}{x}$

And substituting into the DE [A] we get:

$\frac{\frac{dv}{dx}-y}{x}=-\frac{4}{{x}^{2}}-\frac{\frac{v}{x}}{x}+{\left(\frac{v}{x}\right)}^{2}$

$\therefore \frac{dv}{dx}-\frac{v}{x}=-\frac{4}{x}-\frac{v}{x}+\frac{{v}^{2}}{x}$

$\therefore \frac{dv}{dx}=\frac{{v}^{2}-4}{x}$

Which is now separable, so we can collect terms, and "separate the variables" to get:

$\int \frac{1}{{v}^{2}-4}dv=\int \frac{1}{x}dx$

And we can integrate to get:

$\frac{1}{2}{\mathrm{tanh}}^{-1}\left(\frac{v}{2}\right)=\mathrm{ln}x+C$

$\therefore {\mathrm{tanh}}^{-1}\left(\frac{v}{2}\right)=2\mathrm{ln}x+2C$

$\therefore \frac{v}{2}=\mathrm{tanh}(2\mathrm{ln}x+A)$

$\therefore v=2\mathrm{tanh}(2\mathrm{ln}x+A)$

And restoring the substitution:

$xy=2\mathrm{tanh}(2\mathrm{ln}x+A)$

$y=\frac{2}{x}\mathrm{tanh}(2\mathrm{ln}x+A)$

$y\prime =-\frac{4}{{x}^{2}}-\frac{y}{x}+{y}^{2}$.... [A]

This is a non-linear first order Differential Equation. We can attempt a substitution:

$v=xy\iff y=\frac{v}{x}$

And differentiating using the product rule we get:

$\frac{dv}{dx}=x\frac{dy}{dx}+y\Rightarrow \frac{dy}{dx}=\frac{\frac{dv}{dx}-y}{x}$

And substituting into the DE [A] we get:

$\frac{\frac{dv}{dx}-y}{x}=-\frac{4}{{x}^{2}}-\frac{\frac{v}{x}}{x}+{\left(\frac{v}{x}\right)}^{2}$

$\therefore \frac{dv}{dx}-\frac{v}{x}=-\frac{4}{x}-\frac{v}{x}+\frac{{v}^{2}}{x}$

$\therefore \frac{dv}{dx}=\frac{{v}^{2}-4}{x}$

Which is now separable, so we can collect terms, and "separate the variables" to get:

$\int \frac{1}{{v}^{2}-4}dv=\int \frac{1}{x}dx$

And we can integrate to get:

$\frac{1}{2}{\mathrm{tanh}}^{-1}\left(\frac{v}{2}\right)=\mathrm{ln}x+C$

$\therefore {\mathrm{tanh}}^{-1}\left(\frac{v}{2}\right)=2\mathrm{ln}x+2C$

$\therefore \frac{v}{2}=\mathrm{tanh}(2\mathrm{ln}x+A)$

$\therefore v=2\mathrm{tanh}(2\mathrm{ln}x+A)$

And restoring the substitution:

$xy=2\mathrm{tanh}(2\mathrm{ln}x+A)$

$y=\frac{2}{x}\mathrm{tanh}(2\mathrm{ln}x+A)$

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I multiplied the y over and tried to solve it in seperable form (M and N). The partial deritives did not work out to be equal to eachother so I am now stuck finding an integrating factor. Is this the right approach?

${y}^{\prime}=y({y}^{2}-\frac{1}{2})$

I multiplied the y over and tried to solve it in seperable form (M and N). The partial deritives did not work out to be equal to eachother so I am now stuck finding an integrating factor. Is this the right approach?

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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 $

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

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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