Differential Equations: First Order Linear and Separable Equations

Solve the differential equation: $\frac{dy}{dx} + \frac{2x}{1 + x^2} y = 4(1 + x^2)^2$ using an integrating factor.

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.

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

Solve the first-order linear differential equation $y' - 2xy = x$.

Solve the first-order linear differential equation rac{dy}{dx} - 2y = 6 using the integrating factor.

Solve the first-order linear differential equation $y' + \frac{1}{x}y = 6x$ after rewriting it in linear form, using the integrating factor.

y' + 3y = 6

y' + $\frac{y}{x}$ = $e^x$

Convert $xy' + 3y = x$ to $y' + \frac{3}{x}y = 1$

Solve the differential equation $y' + y = e^x$ using the integrating factor $e^{\, \int 1 \, dx}$.

Solve for y in: $x^2 y' + 5xy = x$, converting to $y' + \frac{5}{x}y = \frac{1}{x}$

Solve the differential equation by dividing both sides by $y$ and integrating, then using the transformation $u = \frac{1}{y}$ to solve the equation.

dy/dt = ky

Solve a differential equation of the form $\frac{dy}{dt} = ky$ using separation of variables and integrating both sides to find $y = y_0 e^{kt}$.

Using the factorization technique, solve the differential equation rac{dy}{dx} = y + ky.

Solve the differential equation $\frac{dy}{dt} = ky$.

Given that a lake is affected by an algae bloom covering 1.5 square meters on October 3rd, and grows at a rate proportional to the quantity present covering 4 square meters on October 15th, use a differential equation to find an expression for the area covered at time T. How long will it take to completely cover a lake surface of 5 square kilometers if growth continues in this fashion?