WeBWorK: 6b-Modeling-Cooling and 7-Trench-EulersMethod will be due on Tuesday 3/5 at midnight.
OpenLab: none
"...how it differs from the rocks"
Differential Equations on Math.Stackexchange
- How to solve $(x^2+x)y''+(2-x^2)y'-(2+x)y=x(x+1)^2$I am trying to solve the following $(x^2+x)y''+(2-x^2)y'-(2+x)y=x(x+1)^2$. To find the homogenous solution $y_{h}$ is the part I am having difficulties with, because if would have found $y_{h}$, then I would use variation of parameters to find a particular solution $y_{p}$. To solve $(x^2+x)y''+(2-x^2)y'-(2+x)y=0$ I considered the substitution $$z(x)= ln \ (x) \implies y' = […]
- If $R$ is not simply connected, then the conclusion of Dulac's criterion is no longer valid. Find a counterexample.This is taken from section 7.3 problem 16 of Nonlinear Dynamics and Chaos by Strogatz. If $R$ is not simply connected, then the conclusion of Dulac's criterion is no longer valid. Find a counterexample. I'm having trouble knowing where to start with this. I'm guessing you want your $\nabla \cdot \dot{x} = x^2 + y^2 […]
- deriving differential equation from difference of PDE solutionsConsider defining $$f(x,t)=g(x,t)-h(x,t)$$ where the PDEs describing $g(x,t)$ and $h(x,t)$ are $\it{known}$ but the solutions themselves are $\it{unknown}$. It could be impractical to solve the PDEs for $g(x,t)$ and $h(x,t)$ for whatever reason. But recovering the differential equation for $f(x,t)$ could be useful. I want to determine if a PDE or ODE and be derived […]
- Ordinary Differential Equations and pursuit curvesIn my studies on ordinary differential equations, in the book of title "Equações Diferencias Applicadas" of Djairo Guedes de Figeiredo, I found a problem involving persuit curves: A boat located at the origin proposes to cross a river as shown in the figure enter image description here. Knowing that the speed of the water is […]
- Solve $\frac{dy}{dx}=\frac{2x^2-xy+y^2}{y-x}$.This problem gives me a lot of trouble. I am asked to solve the differential equation $$\frac{dy}{dx}=\frac{2x^2-xy+y^2}{y-x}.$$Note that this is non-linear, non-homogeneous, non-exact, and inseparable for variables $x$ and $y$. This is a problem on my final exam. I gave attempts to test for known form of equations and substitutions but they yield nothing.
- A certain identity involving Tricomi's confluent hypergeometric function.When analyzing one dimensional diffusion processes it is essential to construct the eigen-functions of the infinitesimal generator ${\mathfrak G}_z$ of the process. It turns out that there is always a pair of linearly independent functions both being positive and one being strictly increasing and the other one being strictly decreasing, which span the space of […]
- How to tell which of several possible asymptotic forms a numerical solution to an ODE is converging toI've previously mentioned the ordinary differential equation $$12P\left(f\left(x\right)\right)^3f''''\left(x\right)+12\left(3P-1\right)\left(f\left(x\right)\right)^2f'\left(x\right)f'''\left(x\right)+4\left(3P+2\right)\left(f\left(x\right)\right)^2\left(f''\left(x\right)\right)^2+12\left(P-1\right)f\left(x\right)\left(f'\left(x\right)\right)^2f''\left(x\right)-\frac{9P^2}{Q^4}x = 0$$ where $P$ and $Q$ are control parameters - $P$ is real and positive and $Q$ is real and almost certainly negative. I have boundary conditions $$f\left(1\right) = 1$$ $$f'\left(1\right) = 0$$ $$f''\left(1\right) = 0$$ (and also $f\left(0\right) = 0$, but we'll ignore that one for […]
- Solving a differential equation using separation of variables vs Integrating Factor. Difference in answer.Solving as a variable separable equation: $y'+16xy=6x \rightarrow y'=x(6-16y) \rightarrow \frac{1}{6-16y}dy=x$ integrating we obtain (using u=6-16y, dy=du/-16): $\frac{-1}{16}\ln(6-16y)=\frac{x^2}{2}+C \rightarrow \ln(6-16y)=-8x^2-16C$ and so the solution is: $$y=\frac{3}{8}-\frac{1}{16}e^{-8x^2-16C}$$ But using an integrating factor we have: $y'+16xy=6x \rightarrow e^{8x^2}y'+e^{8x^2}16xy=e^{8x^2}6x \rightarrow [e^{8x^2}y]'=e^{8x^2}6x$. Now integrating we have (using u=$8x^2$,dx=du/16x): $e^{8x^2}y=\frac{3}{8}e^{8x^2}+C$ and the I.F. solution is: $$y=\frac38+Ce^{-8x^2}$$ My answer using the […]
- "Exponent" terminology in the hypergeometric differential equationThe hypergeometric differential equation $$\frac {d^2 f} {d z^2}+ \left( {\frac {1 - \alpha - \alpha'} {z - a} + \frac {1 - \beta - \beta'} {z - b} + \frac {1 - \gamma - \gamma'} {z - c} } \right) \frac {d f} {d z} +\left(\frac{\alpha \alpha^{\prime}(a-b)(a-c)}{z-a}+\frac{\beta \beta^{\prime}(b-c)(b-a)}{z-b}+\frac{\gamma \gamma^{\prime}(c-a)(c-b)}{z-c}\right) \frac{f}{(z-a)(z-b)(z-c)} = 0$$ contains the […]
- Help gaining an Intuitive or Geometric understanding of Linear ODE solutionWorking through Gilbert Strang differential equations and linear algebra to get a deeper understanding of ODEs and their solutions. I came across the following explanation and I cannot understand it: Each input $q(s) ds$ grows to $e^{a(t-s)}q(s) ds$ in the time between $s$ and $t$. Then the solution $y(t)$ comes from these inputs and growth […]
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