Articles of differential equations

Approximation of $\sqrt2$ using Euler's method

Consider the differential equation $$\frac{dx}{dt}=\frac1{2x}.$$ This is a separable O.D.E. so we know how to find all of its solutions: they are of the form $$x(t)=\sqrt{t+C}$$ where $C$ is a constant. Imposing the initial condition $x(1)=1$ fixes $C=0$. Then we have $x(2)=\sqrt2$. Using Euler’s method with $h=1/2$ find an approximation to $\sqrt2$. Provide a numeric […]

Riccati D.E., vertical asymptotes

For the D.E. $$y’=x^2+y^2$$ show that the solution with $y(0) = 0$ has a vertical asymptote at some point $x_0$. Try to find upper and lower bounds for $x_0$: $$y’=x^2+y^2$$ $$x\in \left [ a,b \right ]$$ $$b> a> 0$$ $$a^2+y^2\leq x^2+y^2\leq b^2+y^2$$ $$a^2+y^2\leq y’\leq b^2+y^2$$ $$y’\geq a^2+y^2$$ $$\frac{y}{a^2+y^2}\geq 1$$ $$\int \frac{dy}{a^2+y^2}\geq \int dx=x+c$$ $$\frac{1}{a}\arctan \frac{y}{a}\geq […]

How to solve the differential equation $y' = \frac{x+y}{x-y}$?

Solve the following differential equation: $y’ = \frac{x+y}{x-y}$ Someone please help me start this problem. This does not look like a regular first-order differential equation in the form $y’ + 2xy = x$. Thank you.

How to solve $\frac{dy}{dx}=5xy + \sin x$?

How do I solve $\frac{dy}{dx}=5xy + \sin x$ explicitly? With $y(0) = 1$. I am asked to use an integrating factor. What I did: $\frac{dy}{dx}-5xy = \sin x \\ \text{Integrating factor:} \ e^{\int{-5x\ dx}} = e^{-\frac{5}{2}x^2} \\ \frac{d}{dx}\left[e^{-\frac{5}{2}x^2}y\right] = e^{-\frac{5}{2}x^2}\sin x \\ e^{-\frac{5}{2}x^2}y = \int e^{-\frac{5}{2}x^2}\sin x \ dx$ How would I proceed from there? […]

Show that a differential equation satisfies Lipschitz condition

Prove that if $$\frac{dx}{dt}=(3t^2+1)\cos^2(x)+(t^2-2t)\sin (2x)=f(t,x),$$ then $f(t,x)$ satisfied Lipschitz condition on the strip $S_{\alpha}=\{(t,x):|t|\le\alpha , |x|\le \infty , \alpha >0\}$. Can I get some help for the above problem please.

self similar solution for porous medium equation 2

let $u$ be a function invariant under the scaling transformation $$u_k(x,t)=k^nu(kx,k^{2+(m-1)n}t)$$ $k>0$ such that $$u_k(x,t)=u(x,t)$$ $x\in R^n$ ,$t>0$ , $k>0$ is satisfied . then$u$ can be expressed by $$u(x,t)=t^{-l}w(t^{-l/n}x)$$ with $w(y)=u(y,1)$ $y\in R^n$ $k=t^{-l/n}$ $l=\frac{n}{2+(m-1)n}$ how can we prove that $u$ is a solution of $$\partial_t u-\Delta u^m=0$$ if and only if $w$ satisfies $$\Delta […]

Can someone show me a proof of the general solution for 2nd order homogenous linear differential equations?

I thought I had it figured out but there’s a sort of ‘leap of faith’ at a pivotal point that annoys me. Can someone show me how to derive the general solution of an equation such as: $a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+cy=0$ I want to avoid just saying, “let’s assume the solution is of the form $y=e^{mx}$”. I want […]

Existence of solution to first order ODE. What is there to be proved?

$\bf Theorem$ Let $I$ be an interval of the real line. Let $f(t,x)$ be Lipschitz on $x$ on $I\times \Bbb R$. Let $\tau\in I, \xi\in \Bbb R$. If $\tau$ is interior in $I$, there exists $\lambda>0$ and $x: [\tau-\lambda,\tau+\lambda] \subset I\to \Bbb R$ such that $$x'(t)=f(t,x)\\x(\tau)=\xi$$ Then follows the proof saying that a solution is […]

Verify $y=x^{1/2}Z_{1/3}\left(2x^{3/2}\right)$ is a solution to $y^{\prime\prime}+9xy=0$

This question is a sequel to this previous question. As before, some background information is needed first as follows from my textbook: The standard form of Bessel’s differential equation is $$x^2y^{\prime\prime}+xy^{\prime} + (x^2 – p^2)y=0\tag{1}$$ where $(1)$ has a first solution given by $$\fbox{$J_p(x)=\sum_{n=0}^\infty\frac{(-1)^n}{\Gamma(n+1)\Gamma(n+1+p)}\left(\frac{x}{2}\right)^{2n+p}$}\tag{2}$$ and a second solution given by $$\fbox{$J_{-p}(x)=\sum_{n=0}^\infty\frac{(-1)^n}{\Gamma(n+1)\Gamma(n+1-p)}\left(\frac{x}{2}\right)^{2n-p}$}\tag{3}$$ where $J_p(x)$ is called […]

Thickness of the Boundary Layer

Given an ODE $$\epsilon y”+2xy’=x \cos(x)$$ with boundary condition $y(\pm {\pi \over 2})=2$ Where is the boundary layer and what is the thickness of it?