# Questions tagged [ordinary-differential-equations]

For questions about ordinary differential equations, which are differential equations involving ordinary derivatives of one or more dependent variables with respect to a single independent variables. For questions specifically concerning partial differential equations, use the [tag:pde] instead.

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### How I can solve this nonlinear second order ODE?

I am trying to solve a non-linear ODE. Let $g(x)$ is an arbitrary smooth real function, I want to solve the following equation $2f(x)f''(x) +4(f'(x))^2=g(x)$ for $f(x)$. Multiplying $\dfrac32 f(x)$, ...
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### Simple ODE question. A student learns 150 Chinese characters each month and forgets 10% of what they have learned

A student learns 150 Chinese characters each month and forgets 10% of what they have learned each month as well. Construct a differential equation suitable from the problem Find a solution with a ...
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### $(\beta_n)_{n \geq 0}$ converges uniformly to the solution of $x' = F(t,x)$, variation of Picard iteration?

This is exercise 2.7. from Differential Equations: A Dynamical Systems Approach to Theory and Practice by Marcelo Viana and José Espinar. Let $F \colon \mathcal{U} \to \mathbb{R}^n$ be continuous and ...
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### Is there a way to show that this ode system is asymptotically stable?

Suppose we have $$\dot{x} = -\frac{x}{y+a}$$ $$\dot{y} = -y$$ for $a>0$, Is the above system asymptotically stable? Now, I know that we can solve for $y$ as $$y = y(0) e^{-t}$$ and we can choose ...
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### Can generating functions be used to solve evolution matrix differential equations and recurrence relations of matrices?

Generating functions seem to be a powerful tool in discrete mathematics for solving differential equations and recurrence relations. I've been trying to figure out if these methods can be expanded to ...
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### almost-Newton flows are Newton flows where the chain-rule is 'forgotten', yet its solutions are roots of f anyway, when does this work?

The differential equation for the Newton flow $z (t)$ of $f (t)$ is given by \begin{equation} \dot{z} (t) = - \frac{f (z (t))}{\frac{d}{d t} f (z (t))} = - \frac{f (z (t))}{\dot{f} (z (t)) \dot{z} ...
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### Stability of numerical solution of ODE

I want to solve the ODE \begin{array}{ll} -u''(x)=f(x) & x\in (0,1) \\ u(0)=g(0) \\ u(1)=g(1)\, \\ \end{array} with finite differences using $u''(x)\approx \frac{u(x-h)-2u(x)+u(x+h)}{h^2}$. To ...
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### Reducing a system of ordinary differential equations by imposing relations between the variables

Let's assume that we have the following system of O.D.E. \begin{align} x' = f_1(x,y,z,v),\ y' = f_2(x,y,z,v),\ z' = f_3(x,y,z,v),\ v' = f_4(x,y,z,v) \end{align} for suitable smooth functions $f_1$,...
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### Equation with variable within an integral sign

This seems as a very easy equation but I am having a hard time solving it. I need to solve for $\beta(t)$ . $q_i(t)$'s are given. (Extra fact : $\int q_i(t)^2dt=L_i$ where $L_i$'s are constants) Have ...
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### Defining formulas for first-order linear differential equations.

When defining the formulas for the first-order linear differentiable functions we are necessitated to define a equation that satisfies $u'(x)$ = $u(x)p(x)$ so then the product rule can be applied. And ...
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### Integrating $y' = \sin^2 y$

I'm trying to solve: $$y' = \sin^2 y$$ It is a separable variable differential equation, so I arrive to $$\int\frac{dy}{\sin^2 y}=\int dx$$ I use the identity $$\sin^2 y=\frac{1-\cos(2y)}{1}$$ and ...
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### How to solve $dx/dt=x^3-x$

I want to solve the differential equation: $$\frac{dx}{dt}=x^3-x.$$ If we seek a solution $x$ such that there exist $t_0\in\mathbb R$ verifying $x(t_0)$ equal to $0$, $1$ or $-1$, we find using the ...
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### How to solve $\frac{d^2y}{dx^2} = (x^2 - k_0)y$

What method can be used to solve and plot the y over x graph satisfying: $$\frac{d^2y}{dx^2} = (x^2 - k_0)y$$ for a given constant $k_0$? In particular, I am trying to reproduce the following:
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### Kernel and Differential Operator

I hope your day is great so far. I have a question. I am given a Cauchy-Euler ODE $$\big((x+2)^3\mathbf{D}^3+4(x+2)^2\mathbf{D}^2+3(x+2)\mathbf{D}+1\big)y=\frac{1}{x+2},$$ where $\mathbf{D}^n=d^n/dx^n$...
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### Solving $\frac{d\mathbf{y}}{dt} + \mathbf{G}(t) \mathbf{y} = \mathbf{z}(t)$
Is there a known solution for the linear equation below? $$\frac{d\mathbf{y}}{dt} + \mathbf{G}(t) \mathbf{y} = \mathbf{z}(t)$$ The variables $\mathbf{y}$ is a vector of $n$ elements, $\mathbf{z}$ is ...