Differential Equations

Solve the differential equation $\frac{dy}{dx} - 5y = e^x$ using the integrating factor method.

Solve the differential equation $\sin(x) \frac{dy}{dx} + 3y \cos(x) = \csc(x)$ using the integrating factor method.

Solve the first-order linear ordinary differential equation: $2x \, \cdot \, y' - 3y = 9x^3$ using the integrating factor method.

A tank contains 1000 liters of brine with 15 kilograms of dissolved salt. Pure water is entering the tank at a rate of 10 liters per minute, and the tank drains at the same rate. Determine how much salt is in the tank after 'T' minutes.

Find the derivative of $y$ with respect to $x$.

Differentiate $(2x^2 + 4x - 3)^{199}$ using the Chain Rule.

Draw the slope field for the differential equation $y' = 2y + 3$ and analyze how the slopes change as the value of $y$ changes.

Build a slope field for a given differential equation, using sample points such as (0, 0) and (1, 1) to plot the slopes.

Using a slope field, sketch the solution to a differential equation that passes through a specific point, such as (0, -1).

Find the solution curves by drawing the slope field for the differential equation $\frac{dy}{dx} = x - y$.

Given the differential equation $x^2 + y^2 - 2 = 0$, determine the locations in the graph where the slope ($\frac{dy}{dx}$) is zero, positive, and negative.

Using the given differential equation $x^2 + y^2 - 2 = c$, determine the geometry of isoclines for different values of $c$.

Given the differential equation $\frac{dY}{dX} = Y$ and the initial condition $Y(0) = 1$, use Euler's method with a step size of 1 to approximate $Y(1)$.

Given the differential equation $\frac{dY}{dX} = Y$ and the initial condition $Y(0) = 1$, use Euler's method with a step size of 0.5 to approximate $Y(1)$.

Use Euler's method with a step size of 0.02 to approximate $y(0.08)$ for the given differential equation with initial condition $x_0 = 0, y_0 = 1$.

Estimate the value $Y(4)$ for the initial value problem: $y' - y = 0$ with $Y(0) = 1$ using Euler's method and a step size of 1.

Given that $\frac{dy}{dx} = 3x + y$ and the initial condition $y(0) = -1$, approximate $y(0.2)$ with a step size of $0.04$ using Euler's Method.

Given the differential equation $\frac{dy}{dx} = \frac{x^2 - 4y^2}{8xy + y^4}$, convert it into the standard form of an exact differential equation and find the potential function $F(x, y)$.

Test if the differential equation $3x^2 + 3y^2 \, dx + (3y^2 + 6xy) \, dy = 0$ is exact and solve for the function $F(x, y)$.

Check for exactness: $2xy \, dx + x^2 \, dy = 0$.

Determine if this equation is exact by finding if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}.$