Differential Equations

Given a differential equation with an initial condition $y(0) = 1$ and the differential equation $y' = 3y + 5 \sin(2t)$, find the undetermined coefficients for the particular solution and solve for $C_0$, $C_1$, and $C_2$.

Consider the fourth derivative equation: $y^{(4)} + x^2 y^{(3)} + e^x y = 3$.

Convert the nth order linear differential equation to a system of linear equations in normal form.

Using Euler's method, approximate the solution to the differential equation $y' = f(x, y)$ starting from the point $(x_0, y_0)$ and proceeding with steps $x_1, x_2, \ldots, x_n$.

Given a differential equation $y' = x \cdot y$, approximate the y-value when $x = 1.3$ with an initial condition of $y(1) = 1$ and increment $h = 0.1$.

Given $y' = y$ with an initial condition $y(0) = 1$ and step size $h = 0.01$, use the Euler method to approximate the solution of the differential equation.

Given $y' = y - x^2$ with initial condition $y(0) = 0$ and step size $h = 0.1$, use the Euler method to approximate the solution of the differential equation.

Using substitution methods, solve a first-order differential equation that seems non-separable at first.

Given a third order differential equation, reduce it to a system of three first order differential equations using variables $X_1$, $X_2$, and $X_3$ where $X_1 = x$, $X_2 = x'$, and $X_3 = x''$.

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

Given the second order differential equation $y y' = 6 (y')^2$, solve by making the substitution $y' = \frac{dp}{dx}$ and finding the general solution.

Find the general solution and a second solution of the differential equation $x^2 y'' + 5x y' - 5y = 0$ given that $y_1 = x$ is a known solution.

Given $t²$ is a solution to $t^2y'' + 3ty' - 8y = 0$ Solve using the reduction of orders method.

Use the reduction of order formula to find another solution to this differential equation: $9y'' - 12y' + 4y = 0$.

Find the general solution to the differential equation: $\frac{d^2y}{dx^2} + 4 \frac{dy}{dx} + 4y = 0$.

Derive the general form solution for a second-order constant coefficient linear homogeneous differential equation with repeated roots.

Solve the second-order linear constant coefficient homogeneous equation $y'' - 2y' + y = 0$ using the reduction of order method to find a second independent solution.

Find the general solution to the second order linear homogeneous differential equation $y'' + 2y' + y = 0$.

Given a second order linear homogeneous differential equation with a repeated root, find the general solution.

Solve the second order homogeneous linear differential equation $4y'' - 5y' - 6y = 0$ using the method of characteristic equations.