Differential Equations

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.

Given a differential equation, find the two linearly independent solutions and show that they form a fundamental set of solutions using the Wronskian.

Using variation of parameters, solve the differential equation: $y'' + y = \sec(x)$ on the domain from $0$ to $\frac{\pi}{2}$.

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 a system of linear first-order differential equations using matrix methods.

Use the Matrix method to solve this linear system of differential equations: rac{dX}{dt} = 6x + 5y and rac{dY}{dt} = x + 2y.

Rewrite the second order differential equation as a system of first-order linear differential equations.

Rewrite the fourth order differential equation as a system of first-order linear differential equations.

Solve the system by transforming it into a single differential equation: $X' = -2Y, Y' = X$.

Solve the system by transforming it into a single differential equation: $X' = Y, Y' = 6X - Y$.