📘 differential equations

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

1. **Problemstellung:** Gegeben sind die Differentialgleichungen:

1. **State the problem:** Solve the initial-value problem $$y' = 3e^{3x - y}, \quad y(0) = \ln(4).$$ 2. **Rewrite the differential equation:** We have $$\frac{dy}{dx} = 3e^{3x - y}

1. **State the problem:** Solve the initial-value problem $$\frac{dy}{dt} = \frac{4y}{t}, \quad y(2) = 3.$$\n\n2. **Identify the type of differential equation:** This is a first-or

1. **State the problem:** Solve the initial-value problem $$\frac{dy}{dx} = \frac{xy}{x+3}, \quad y(-2) = 3,$$ assuming $x > -3$. 2. **Rewrite the differential equation:** Separate

1. **State the problem:** We want to find values of $k$ such that the function $y = 5 \cos(kt)$ satisfies the differential equation $$16 \frac{d^2 y}{dt^2} = -4y.$$\n\n2. **Write t

1. **State the problem:** We want to find the values of $r$ such that the function $y = e^{r x}$ satisfies the differential equation $$\frac{d^2 y}{dx^2} - 6 \frac{dy}{dx} + 8 y =

1. **State the problem:** Solve the system of differential equations: $$\begin{cases} x' = -4x \\ y' = -6x - y \end{cases}$$

1. **Problem statement:** We are given a slope field for a differential equation and asked to find which particular solution satisfies the initial condition $y(0) = 0$. 2. **Step 1

1. **State the problem:** Solve the differential equation $$\frac{dy}{dp} = B(6400y - y^2)$$ where $B$ is a constant. 2. **Rewrite the equation:** Factor the right side:

1. **State the problem:** We have the differential equation $$y' = y + 3xy^{5}$$ with initial condition $$y(0) = -1$$.

1. Il problema di Cauchy dato è: $$y' - 2y = xe^{x^2+2x}, \quad y(0) = \frac{3}{2}.$$

1. **Problem statement:** We want to find the two times $t$ when the water height $h(t)$ is 40 cm. 2. **Given:** The differential equation is

1. **Problem statement:** We are given the differential equation $$\frac{dy}{dx} = \sin x$$ with initial condition $$y(0) = 0$$. We want to estimate $$y(1)$$ using Euler's Method w

1. **State the question:** Why do we multiply both sides of the differential equation by the integrating factor? 2. **Recall the original equation:** $y' + P(x) y = Q(x)$.

1. **State the problem:** Solve the first-order linear differential equation $$y' + e^{2x} y = e^x.$$\n\n2. **Identify the type and formula:** This is a linear ODE of the form $$y'

1. **State the problem:** Solve the first-order linear differential equation $y' + e^x y = e^x$. 2. **Identify the type and formula:** This is a linear differential equation of the

1. **State the problem:** Solve the differential equation $y' + 2xy = x$. 2. **Identify the type:** This is a first-order linear ordinary differential equation of the form $y' + P(