differential equations – Piecewise function and NDSolve

I’m trying to solve a system of differenial equations involving a piecewise function with NDSolve. I tried to define this function using Piecewise but the kernel keeps crashing.
I also tried to express the function in terms of UnitStep and HeavisideThetaby means of Simplify`PWToUnitStep but the crash still takes place.
(More details on the system of equations and its numerical implementation can be found here Kernel crash after repeated use of NDSolve)

Do you know if there is any other alternative or workaround that I could use in order to avoid this annoying crash?

Thank you!!

partial differential equations – Solve $u_t$ + -$u__xx$ + $xu$ using Fourier Transform

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partial differential equations – Solution to PDEs and inequalities

I am working on a modelling problem, and I have derived the following result about two functions $f(x,y)$ and $g(x,y)$ that are continuous and differentiable in both the arguments–

$$frac{frac{partial}{partial x} f(x,y)}{f(x,y)} – frac{frac{partial}{partial x} g(x,y)}{g(x,y)} = h(x)<0$$

$$frac{frac{partial}{partial y} f(x,y)}{f(x,y)} – frac{frac{partial}{partial y} g(x,y)}{g(x,y)}<0$$

These appear to be partial derivatives of the natural log of the functions, so I tried some exponential functions, but in all the tries I made,whenever the first held true, the second turned 0.

I am new to PDEs, so I may be missing some very basic thing.

I would appreciate every answer, either solving it, providing resources to study to be able to myself solve it (given I am new to PDEs) or at least giving examples of functions that satisfy these conditions if not the general solution.

Help proving one direction of an iff for a self learner in differential equations

I am self learning differential equations from the book “Differential Equations With Hostorical Applications” by George Simmons. The following problems is the one I am having an issue with:

Given the homogeneous equation $y” + P(x)y’ + Q(x)y =0$, and change the independent variable from $x$ to $z=z(x)$. Show that the homogeneous equation can be transformed through this change of variables into an equation with constant coefficients iff $frac{Q’ + 2PQ}{Q^{3/2}}$ is constant, in which case $z = int sqrt{Q(x)}dx$ will effect the desired result.

As of now I have solved the “only if” direction with the following math:

Let $ z = z(x)$. We have the following:

$$frac{df}{dx} = frac{df}{dz}frac{dz}{dx} = z'(x) frac{df}{dx} Rightarrow frac{d}{dx} rightarrow z'(x)frac{d}{dz}$$

for the first derivative, and for the second derivative we have:

$$frac{d^{2}f}{dx^{2}} = frac{d}{dx}left(z'(x)frac{df}{dz}right) = frac{d}{dz}left(z'(x)frac{df}{dz}right)z'(x) = (z'(x))^{2}frac{d^{2}f}{dz^{2}} + z'(x)frac{d}{dz}left(z'(x)right) frac{df}{dz} = ldots $$
$$ ldots = (z'(x))^{2}frac{d^{2}f}{dz^{2}} +z'(x)frac{d}{dx}left(z'(x)right)frac{dx}{dz} frac{df}{dz}$$

$$frac{dx}{dz} = frac{d}{dz}left(z^{-1}(z) right) = frac{1}{z'(x)}$$
$$Downarrow$$
$$frac{d^{2}f}{dx^{2}} = (z'(x))^{2}frac{d^{2}f}{dz^{2}} + z”(x)frac{df}{dz} Rightarrow frac{d^{2}}{dx^{2}} rightarrow (z'(x))^{2}frac{d^{2}}{dz^{2}} + z”(x)frac{d}{dz}$$

and all together we have the following three equations, which combined gives us the transformed differential equation.

$$y” + P(x)y’ + Q(x)y =0$$

$$frac{d}{dx} rightarrow z'(x)frac{d}{dz}$$

$$frac{d^{2}}{dx^{2}} rightarrow (z'(x))^{2}frac{d^{2}}{dz^{2}} + z”(x)frac{d}{dz}$$

$$y” + left(frac{P(z)}{z'(x)} + frac{z”(x)}{(z'(x))^{2}}right)y’ + frac{Q(z)}{(z'(x))^{2}}y = 0$$

Now suppose that $frac{Q(z)}{(z'(x))^{2}} = c_{2}$ and that $left(frac{P(z)}{z'(x)} + frac{z”(x)}{(z'(x))^{2}}right) = c_{1}$. Plug $z'(x) = sqrt{frac{c_{2}}{Q(z)}}$ and $z”(x) = frac{Q'(x)}{2sqrt{c_{2}Q(x)}}$ into the the equation for $c_{1}$, and we get that :

$$frac{2PQ + Q’}{Q^{3/2}} = frac{2c_{1}}{sqrt{c_{2}}} = constant$$

I have tried various things for the other direction, but I can’t seem to make any progress.

symbolic – High-order Differential Equation

I need help in solving the differential equation below. It would be nice if someone can help me out or guide how can I proceed. I have looked into many references, but I could not find something similar to the problem I have.
$$ ddddot{x} + frac{w_1}{w_2}ddot{x}+frac{1}{w_2}lambda(t) =0 $$

where, $$ lambda(t)= -c_1cdot t + c_2 $$ and $$w_1, w_2, c_1, c_2text{ are constants.}$$

Thanks in advance.
PS- I need to solve this analytically.

ordinary differential equations – Find integrating factor

I am trying to solve this equation but I can’t really find an integrating factor:
$$underbrace{ycdot(1+x)}_{P}:dx+underbrace{xcdot(1+y)}_{Q}:dy=0$$
I know I must find $mu equiv mu (x, y)$ so that:
$$frac{partial}{partial y}(mu P)=frac{partial}{partial x}(mu Q)$$
Therefore:
$$Pfrac{partial mu}{partial y}+mu frac{partial P}{partial y}=Qfrac{partial mu}{partial x} + mu frac{partial Q}{partial x}$$
I have trouble working out the partials of $mu$. I have tried doing $muequiv mu (epsilon)$ and $epsilon equiv epsilon (x, y)$, therefore:
$$frac{partial mu}{partial y}=frac{partial mu}{partial epsilon} cdot frac{partial epsilon}{partial y}$$
And the same for the $x$. But I don’t know where to go from here. Could someone please help me?

ct.category theory – Examples of connection preserving maps in differential geometry

In synthetic differential geometry and tangent categories, linear connections on the tangent bundle are treated as a sort of algebraic gadgets that incorporate the tangent bundle. Like any other algebraic gadget, it seems natural to consider morphisms $f:M to N$ that preserve a chosen linear connection $(M, nabla), (N, nabla’)$. I’ve tried searching through the differential geometry literature, and connection-preserving morphisms doesn’t seem to be discussed very much (except the case where an isometry between Riemannian manifolds induces a connection-preserving morphism between their Levi-Civita connections).

Have connection-preserving maps been considered by differential geometers, and if so, can someone point me in that direction?

differential equations – How do you find the Inverse of Elliptic İntegral of Second Kind when modulus is large

So I tried to take the inverse of EllipticE when modulus is large, in Mathematica, but the solution gives wrong answer.

InverseSeries(Series(EllipticE(x, -k), {x, 0, 12}, {k, Infinity, 1}),y) = InverseFunction(y,k)

For example, I tried EllipticE(0.5,-9.9) = 0.656 where x:0.5 , k:-9.9, y:0.656

But InverseFunction(y,k) is not equal to 0.5. Am I not correctly taking the inverse of the function?
I need a general form of an equation for the inverse of EllipticE. Polynomial approximation is also fine. The approximation should definitely work around when x-->0 and k-->-infinity.
So for the above example, the approximation function result should yield to 0.5 when y=0.656 and k=-9.9. I need to code this function in MCU, so I need an analytical approximation.

Caner

plotting – How To Generate a Direction Field and Solve a System of Differential Equation

I want to draw a direction field and solve this system of differential equations using Mathematica but I’ve been researching, and I can’t find a way to do this.

$$mathbf{x’} = begin{pmatrix} 1 & 1\ 4 & -2 end{pmatrix} mathbf{x}$$

Note: I’m still a newbie to Mathematica so please be patient with me. Thanks!

linear algebra – Non-Homogenous Differential Equations

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