The reasoning behind variation of parameters. November 29    2

Let's say you have the second order equation: $y''+p(x)y'+q(x)y=f(x)$And let's say you have found two solutions ($y_1$ and $y_2$) to the homogeneous equation:$y''+p(x)y'+q(x)y=0$.Then the method using variation of parameters says that we should try a

Estimate the variation of solution in ODE November 29    1

For the initial value problem, $$y' = x + e^xsin(xy), \; y(0) = 0 =y_0$$ estimate the variation of the solution in the interval [0,1] if $y_0$ is perturbed by 0.01. The way I use to approach this problem is let $y_0 = 0.001, y_1 = 0$, then we can use

Find a closed form of $\sum_{i=1}^n i^3$ November 29    3

Find a closed form of $\sum_{i=1}^n i^3$
I'm trying to compute the general formula for $\sum_{i=1}^ni^3$. My math instructor said that we should do this by starting with a grid of $n^2$ squares like so: $$ \begin{matrix} 1^2 & ...

Variational differentional equations November 29    1

For $f \in C^1(D)$, $D$ compact, there exists unique solutions (locally) for$\dot{y} = f(t,y)$, $y(t_0) = y_0$.We denote the solution with $y(t;t_0,y_0)$.Let $G(t;t_0,y_0) := \frac{\partial}{\partial y_0} y (t;t_0,y_0)$ and $g(t;t_0,y_0) := \frac{\pa

Examples of solutions of an ODE with this property November 29    1

Examples of solutions of an ODE with this property
Let $f:\mathbb R\times \mathbb R^n\to \mathbb R^n$ be a $C^1$ class function and suppose that $\varphi(t)$ defined in $\mathbb R$ is a solution of $x'=f(t,x)$, $x(t_0)=x_0$.I've been thinkin ...

How do I solve the DE $y''+6y'+8y = 2t+e^t$ November 29    1

The differential equation I am trying to solve is $$ \dfrac{d^2y}{dt^2} + 6\dfrac{dy}{dt} + 8y = 2t +e^t $$ I know how to start off. I have done the $s^2 + 6s + 8 = 0$ to get $s = -4$ and $s = -2$ and have the $$ y_p(t) = k_1e^{-4t} + k_2e^{-2t} $$Wh

If $g\lt f$, then $\varphi(t)\lt \psi(t)$ November 29    1

I'm trying to solve this question:In the rectangle $P=\{(t,x);|t-t_0|\lt a,|x-x_0|\lt b\}\subset \mathbb R^2$, let $f,g$ be two continuous functions and locally Lipschitz. If $g\lt f$ in P, then for $\varphi$ and $\psi$ solutions of, respectively, $x

Closed form of an integral November 29    1

Is there a closed form of $$\int\limits_0^1\frac{\arctan(\sqrt{x^2+2})}{(1+x^2)\cdot(\sqrt{x^2+2})}dx \quad \text{?}$$I just know that $\int\limits_0^1\frac{\arctan(\sqrt{x^2+2})}{(1+x^2)\cdot(\sqrt{x^2+2})}dx = 0.514042...$This is called the Ahmed i

You Might Also Like