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Reduce ODE into 1st order

I have an explicit set of differential equations: $\ddot{x}=f(x,\dot{x})$ I would like to reduce it in the following way: $\dot{y} = g(y)$ by substitutions as shown here: wikipedia. I have done ...
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What is the correct way to input these boundary conditions for the following nonstandard Laplace equation, whose coefficients of $\frac{\partial^2 u}{\partial x^2}$ and $\frac{\partial^2 u}{\partial y^... • 11.9k 12 votes 4 answers 1k views Position of discontinuous coefficient influences the solution of PDE This issue is raised in the discussion under this post about heat flux continuity and I think it's better to start a new question to state it in a clearer way. Just consider the following example: <... • 66.2k 8 votes 2 answers 628 views DirichletCondition in NDSolve causes asymmetry (Finite Element Method) Edit Thanks for the responses. I now agree with xzczd and Bill Watts who kindly gave answers and user21 who wrote useful comments. Both my cases below are correct. On reflection I think I made an ... • 16.5k 3 votes 1 answer 645 views How to solve fourth order differential equation? I have a differential equation of this type: y[x] - 1 - 2*l^2*y''[x] + l^4*y''''[x] == 0 (where l is a parameter and ... 2 votes 2 answers 814 views What causes the warning "The dependent variable in the boundary condition needs to be linear" when using NDSolve? V 12.1.1 on windows 10 Why the following works ... • 144k 7 votes 1 answer 306 views 1D-waveequation with absorbing boundary condition: FEM solution? I try to simulate the special absorbing(?) boundary condition Derivative[1, 0][y][1, t] + Derivative[0, 1 ][y][1, t] == 0 which only allows energy flow in ... 3 votes 1 answer 389 views NDSolve 1D Heat equation with NeumannValue poorly satisfies boundary condition Context I am trying to measure the slope of the stellar cuspide around our MilkyWay's central black hole using the observed distribution of S stars orbiting around it. Attempt Close to the ... • 23k 5 votes 1 answer 217 views Problem with using NDSolve on wave PDE on string I am trying to verify my hand solution for wave PDE in 1D, on a string. The solution I get from NDSolve does not match the analytical solution. This is a string, ... • 144k 2 votes 1 answer 185 views How to solve Neumann boundary ODE with shooting method?$\rho(x)$is the distribution function. I'm trying to solve the following equation:$D_t\frac{\partial^2\rho}{\partial x^2}-\frac{\partial (v_0\rho)}{\partial x}=0D_t\frac{\partial\rho}{\partial x}-...
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While trying to get familiar with the NeumannValue function I tried solving the wave equation on $r\in [0,2\pi]$ for $\theta(t,r)$ $$\partial^2_t \theta - \partial^2_r \theta =0,$$ with intial ...