📘 differential equations

1. **Problem:** Solve the differential equation $$\frac{d^2 x}{dt^2} - 2 \frac{dx}{dt} + 5x = 34 \cos 2t$$ with initial conditions $$x(0) = 10$$ and $$\frac{dx}{dt}(0) = 2$$. 2. **

1. The problem is to solve the differential equation $$\frac{dy}{dt} = (ty)^2$$. 2. This is a separable differential equation because the right side can be written as a product of

1. Masalah pertama: Tentukan solusi persamaan diferensial homogen $$\frac{d^2y}{dt^2} + 3\frac{dy}{dt} - 10y = 0$$ dengan kondisi awal $$y(0) = 1$$ dan $$y'(0) = 3$$. 2. Gunakan tr

1. Diketahui persamaan diferensial orde kedua nonhomogen: $$\frac{d^2 y}{dx^2} - 8 \frac{dy}{dx} + 16y = 24 e^{4x}$$

1. **Menentukan solusi homogen $y_h$** Persamaan diferensial homogen terkait adalah:

1. **Stating the problem:** We are given the nonhomogeneous differential equation:

1. **State the problem:** Determine the properties of the differential equation $$y'' + ty' + 4y = 0$$ from the given options. 2. **Recall definitions:**

1. **State the problem:** We are given the differential equation $$ty' + 2y = 4t^2$$ with the initial condition $$y(1) = 2$$.

1. **State the problem:** Solve the differential equation $$\frac{d^2x}{dt^2} - 2\frac{dx}{dt} = e^t (t - 3)$$ with initial conditions $$x(0) = 2$$ and $$x'(0) = 0$$. 2. **Identify

1. **State the problem:** We want to find the general solution of the system of differential equations $$\mathbf{y}' = A\mathbf{y}$$ where $$A = \begin{bmatrix}-1 & 1 & 0 \\ 0 & -1

1. **State the problem:** Solve the differential equation $y(4) + 2y + y = 0$. 2. **Clarify notation:** Here, $y(4)$ means the fourth derivative of $y$ with respect to $x$, so the

1. **State the problem:** Solve the differential equation $$2xy - 9x^2 + (2y + x^2 + 1) \frac{dy}{dx} = 0.$$\n\n2. **Rewrite the equation:** Rearrange to isolate $$\frac{dy}{dx}$$:

1. **Problem statement:** Find the power series solution of the differential equation $$xy'' + y' + xy = 1$$ up to the term in $x^5$, given initial conditions $y(0) = 1$ and $y'(0)

1. **State the problem:** Solve the differential equation $$x^2 y'' + x y' + y = \sin(\ln(x))$$ where $y' = \frac{dy}{dx}$ and $y'' = \frac{d^2 y}{dx^2}$. 2. **Recognize the type:*

1. The problem is to rewrite the differential equation $$\frac{dx}{dt} + t^3 x = \sin(t) x^4$$ after a proper substitution. 2. This is a Bernoulli differential equation of the form

1. **State the problem:** Solve the second-order linear differential equation $$\frac{d^2 x}{dt^2} - \frac{dx}{dt} - 6x = 0.$$\n\n2. **Identify the type of equation:** This is a ho

1. The problem is to identify the correct differential operator $L$ that represents the given differential equation: $$4 \frac{d^2 x}{dt^2} = \sin(t) \frac{dx}{dt} + 3t \cdot x = t

1. The problem is to find the characteristic equation of the differential equation $$2 \frac{d^3X}{dt^3} + 4 \frac{dX}{dt} - 3X = 0$$. 2. For linear differential equations with con

1. The problem is to identify two correct statements about the differential equation $$\frac{dX}{dt} + e^{2t} \cdot X = t^2$$. 2. First, identify the dependent and independent vari

1. **State the problem:** Solve the differential equation $$t^2 \frac{dx}{dt} + 2tx - 3t^2 = 0$$ for $x$ as a function of $t$. 2. **Rewrite the equation:** Divide both sides by $t^

1. The problem is to analyze the differential equation $$t^2 \frac{dx}{dt} + 2tx - 4t^3 = 0$$ and find correct statements about it. 2. First, rewrite the equation in standard form