Differential Equations: Exact and Bernoulli Equations

Solve a non-exact differential equation using an integrating factor given the equation: $3x^2 y + x y^2 + x^3 + x^2 y$.

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 $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}.$

Check for exactness: $4x \sin y \, dx + 2x^2 \cos y \, dy = 0$. Determine if this equation is exact by finding if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.

Check for exactness: $3x^2 y \, dx + x^3 y \, dy = 0$.

Determine if this equation is exact by finding if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$, and adjust using an integrating factor if necessary.

Check for exactness: $\displaystyle (2x+4y-1) \, dx + (4x+6y+1) \, dy = 0$. Determine if this equation is exact by finding if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ and find the general solution of the exact differential equation.

Solve the exact equation using the given partial derivatives.

Solve the Bernoulli differential equation: $y' - 5y = -\frac{5}{2}x y^3$ using the substitution $u = y^{-2}$ and the method of integrating factors.

Using the Bernoulli equation, solve the differential equation of the form: $y' + P(t)y = G(t)y^n$ by introducing a new variable $V$ and applying the integrating factor method.

For the Bernoulli differential equation $\frac{dy}{dx} + p(x)y = q(x) y^n$, manipulate it to get rid of the $y^n$ term by multiplying both sides by $y^{-n}$, and solve for $y$ using the substitution $u = y^{1-n}$.

Solve the Bernoulli differential equation $\frac{dy}{dx} + p(x) y = q(x) y^2$ by multiplying both sides by $y^{-2}$ and performing a variable substitution.

Solve the Bernoulli differential equation: rac{dy}{dx} - y = e^{2x} y^3

Verify that these two are integration factors for this equation and use them to solve the equation.