13. Intro to Differential Equations

Solve the differential equation $e^{x}y\frac{dy}{dx}=e^{-y}+e^{-3x}-y$ by separation of variables.

Solve the differential equation $x\frac{dy}{dx}=6y$ by separation of variables.

Which of the following is the solution to the differential equation $\frac{dy}{dx}=4xy$, where $y\left(2\right)=-2$?

Solve the following system of differential equations by systematic elimination:

$$\frac{\partial x}{\partial t}=2x-y$$$$\frac{\partial y}{\partial t}=x$$
Which of the following gives the general solution for $x\left(t\right)$?

Which of the following is the general solution to the differential equation $\frac{dy}{dx}=9x^2{y}^2$?

Integral Equations

In Exercises 7–12, write an equivalent first-order differential equation

and initial condition for y.

y = ln x + ∫ₓᵉ √ (t² + (y(t))²) dt

33–42. Solving initial value problems Solve the following initial value problems.

p'(x) = 2/(x² + x), p(1) = 0

Consider the following differential equation: $x\frac{dy}{dx}-y=x^2\sin \left(x\right)$. Which of the following is the correct integrating factor to solve this equation?

Find the general solution of the differential equation: $y^{\left(4\right)}-2y^{\left(2\right)}+y=0$.

Solve the differential equation $y^{\mathrm{\text{'}\text{'}}}+3y^{\mathrm{\text{'}}}+2y=cos\left(ex\right)$ using the method of variation of parameters. Which of the following is the general solution?

Which of the following represents the general solution to the differential equation $\frac{dy}{dx}=\frac{1}{x}\left(\ln 2\right)$?

Solve the differential equation by separation of variables: $\frac{dp}{dt}=p-p^2$. Which of the following is the general solution?

21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.

p'(x) = 16/x⁹ - 5 + 14x⁶

Solve the differential equation $5y″-10y\prime +10y=e^{x}secx$ using the method of variation of parameters. Which of the following is the general solution?

State the order of the differential equation and indicate if it is linear or nonlinear. 

$y^{{}^{\prime \prime \prime }}+3xy=4\sqrt{x}$