WeBWorK: Assignments 7-SecondOrderRepeated, and 8-SecondOrderComplex due Tuesday, 3/26, at midnight.
OpenLab: Your Project is due April 4 (Part 1, Example Data, Part 2)
"...how it differs from the rocks"
Differential Equations on Math.Stackexchange
- Finding integral curves to IVPs with accumulation functionsI find it rather elegant that if you have a separable, first-order initial-value problem, $$\frac{dy}{dx} = \frac{f(x)}{g(y)}, \qquad y(a) = b$$ then you can implicitly solve it simply by creating two definite integrals (i.e. accumulations functions) based on the initial conditions: $$\int_{b}^{y} g(s)\,ds = \int_{a}^{x} f(t)\,dt$$ In particular, this means that any graphing software which […]
- Separation of variables issue [duplicate]I'm currently trying to solve the following PDE - $$ \partial_{t}u(x,t) - \partial_{xx}u(x,t) = 0 \quad 0
- Especial second order differential equation $y^{\prime\prime}y^{\prime}=f(x,y,y^{\prime}).$Can anyone provide examples of physical situations or mathematical problems where differential equations of the form $$y^{\prime}y^{\prime\prime}={A{y^{\prime}}^{5}+B{y^{\prime}}^{ 4}+C{y^{\prime}}^{3}+D{y^{\prime}}^{2}+Ey^{\prime}+F},$$ were $A,B,C,D,E$ are functions of $(x,y)$ only are useful ?
- Will solutions of SDE with different initializations intersect at some point?Inspired by the result for ODE here, which shows that solutions to the same ODE with different initializations do not intersect, I am wondering if similar results also hold for SDE? Consider the SDE $$dX_t=f(X_t,t)dt+g(X_t,t)dB_t,$$ if both $f,g$ are Lipschitz with linear growth, then there must exists a unique strong solution for any independent initialization. […]
- Jacobian matrix from mass action kinetic ODE networkI have an ODE system consisting of >1000 equations for which I would like to obtain the Jacobian matrix. For simplicity, let's consider the following example: X1 --> X2 + X3 (on1) X1 + X2 --> X1 (off1) X3 --> X4 + X5 (on2) X4 + X5 --> X3 (off2) Assuming mass-action kinetics, the derivative […]
- Particular integral for nth order differential equations with constant coefficientsSay, I have an nth order differential equation with constant coefficients of the form, $$f(D)y=w(x)$$ Where $f(D)=\sum_{k=0}^{n}a_{k}D^{k}$ And $D^{k}$ denotes the $k$th derivative and $w$ is complete function in $x$. I need to show that solutions to such equations are of the form, $$y=p(x)+c(x)........(1)$$ Where $p(x)$ is particular integral and $c(x)$ is complementary function. I […]
- Confluent-like Hypergeometric FunctionI'm trying to prove that one of the solutions of the equation $$xy''+y'+(c-x)y=0$$ is a Confluent Hypergeometric Function of the Second Kind times an exponential, that's what Mathematica says, I don't know how should I start
- Intuition behind dampening as a rotation in the complex planeI'm currently building my intuition behind the meaning of the variables within the general second-order damped harmonic oscillator frequently taught in engineering: $$ \ddot{x} + 2\zeta\omega_n\dot{x} + \omega_n^2 x = 0 $$ Assuming an underdamped case, the eigenvalues are as follows: $$ \lambda = -\zeta w_n \pm w_n \sqrt{1 - \zeta^2} i $$ As such, […]
- Differential equation LagrangeI have a problem solving this differential equation: $$ y = x(y')^2 - y' $$ Is anyone able to work out for me how to do this because I am at a loss? I would be very grateful for your help! Thanks.
- Prove that two integrating factors define a solution.I've been toiling away at this proof problem from Chapter 2.4 ending exercises of Differential Equations 3rd ed by Shepley L. Ross, but to no avail. Show that if $\mu (x, y)$ and $v(x, y)$ are integrating factors of $$ M(x, y)\ dx + N(x, y)\ dy = 0 $$ such that $\mu (x, y) […]
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