📘 differential equations

1. **Problem statement:** Find the eigenfunction expansion of $f(x) = x$ using the eigenfunctions $y_n(x) = \sin\left(\frac{n\pi x}{L}\right)$ from the Sturm-Liouville problem $y''

1. The problem is to understand how the boundary conditions $X'(0)=0$ and $X'(\frac{\pi}{2})=0$ lead to the general solution for $X(x)$. 2. The general form of the solution to a se

1. **State the problem:** Solve the differential equation $$x(x+y)\frac{dy}{dx} - y(3x+y) = 0$$ using the substitution $$y = vx$$. 2. **Substitution:** Let $$y = vx$$, so $$\frac{d

1. **State the problem:** Solve the differential equation $$y' = 1 + y^2 + \sin x$$ with the initial condition $$y(0) = 0$$. 2. **Understand the equation:** This is a first-order n

1. **State the problem:** Solve the differential equation with the initial condition $y(0)=0$. 2. **Identify the differential equation:** Since the user did not specify the equatio

1. **State the problem:** We need to solve the differential equation $$y' = (1 + y^2) + \sin x$$ where $y'$ denotes the derivative of $y$ with respect to $x$. 2. **Rewrite the equa

1. **State the problem:** Solve the differential equation $$y'' + 4y = 5t^2 e^t$$ for the general and particular solutions. 2. **Identify the type of equation:** This is a nonhomog

1. **Problem 5:** Find the particular solution to the differential equation $$y\sqrt{1 - x^2} y' - x\sqrt{1 - y^2} = 0$$ with initial condition $$y(0) = 1$$. 2. **Rewrite the equat

1. **Stating the problem:** Solve the differential equation given (though the exact equation is not specified, we assume a common form such as $\frac{dy}{dx} = f(x,y)$). 2. **Gener

1. **State the problem:** We are given the differential equation $$y'(t) = \frac{1}{t} y(t) - y(t)^2$$ and the substitution $$y(t) = \frac{1}{z(t)}.$$ We want to find an equation f

1. **Problem statement:** Given the differential equation $$\frac{d^2x}{dt^2} + 5 \frac{dx}{dt} + 6x = 0,$$ we are to rewrite it as a system of first order differential equations,

1. **State the problem:** We are given the second-order differential equation $$\frac{d^2x}{dt^2} - 9 = 4t$$ with initial conditions $$x=0$$ and $$t=0$$.

1. **Problem statement:** Solve the differential equation $$y'' - y = f(t)$$ where $$f(t) = |t|$$ for $$-\pi \leq t \leq \pi$$ and $$f(t)$$ is periodic with period $$2\pi$$ using c

1. **Problem Statement:** Solve the differential equation $$y'' - y = f(t)$$ where $$f(t) = \begin{cases} 1, & 0 < t < \pi \\ 0, & \pi < t < 2\pi \end{cases}$$. 2. **Step 1: Unders

1. The problem involves finding the particular solution $y_p(t)$ given by the series: $$y_p(t) = \frac{\pi}{8} - \frac{2}{\pi} \sum_{n=-\infty}^\infty \frac{e^{i(2n-1)t}}{(2n-1)^2

1. **Problem statement:** Solve the differential equations with periodic forcing function $f(t) = |t|$ for $-\pi \leq t \leq \pi$ and $f(t) = f(t + 2\pi)$. 4. Solve $$y'' - y = f(t

1. Problemstellung: Wir sollen das Anfangswertproblem $$y'(t) = y(t) + t^2$$ mit der Anfangsbedingung $$y(0) = 1$$ lösen. 2. Formel und Methode: Dies ist eine lineare Differentialg

1. **State the problem:** We have a leaking water tank with volume $V$ at time $t$ hours, and the rate of change of volume is proportional to the volume itself: $$\frac{dV}{dt} = -

1. **Stating the problem:** Solve the differential equation:

1. **State the problem:** Find the general solution of the differential equation $$y'' + 7y = 0$$. 2. **Characteristic equation:** Assume a solution of the form $$y = e^{rt}$$. Sub

1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} = 2y$$ and find the general solution. 2. **Use the method of separation of variables:** Rewrite the equati