13. Intro to Differential Equations

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The differential equation y′+2y=t is first-order, linear, and separable.

43–44. Motion in a gravitational field: An object is fired vertically upward with initial velocity v(0)=v₀ from initial position s(0)=s₀.

a. For the following values of v₀ and s₀, find the position and velocity functions for all times at which the object is above the ground (s = 0).

v₀ = 49 m/s, s₀ = 60 m

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

tan θ dr/dθ + r = sin²θ, 0 < θ < π/2

Given the linear system of differential equations $$\begin{array}{l}\frac{\partial x}{\partial t}=2x+y\\ \frac{\partial y}{\partial t}=x+2y\end{array}$$find the most general real-valued solution.

Solve the differential equation using variation of parameters: $y^{\mathrm{\text{'}\text{'}}}+3y^{\mathrm{\text{'}}}+2y=17+e^x$. Which of the following is the general solution?

Which of the following functions is a solution to the differential equation $4y+y^{″}=0$?

Solve the differential equation using variation of parameters: $y^{\mathrm{\text{'}\text{'}}}+3y^{\mathrm{\text{'}}}+2y=\frac{1}{3}+e^x$. Which of the following is the general solution?

Solve the differential equation $e^{x}y\frac{dy}{dx}=e^{-y}+e^{-5x}-y$ by separation of variables.

Solve the differential equation by separation of variables: $\frac{dy}{dx}=\frac{2y+5}{{8x+9}^2}$. Which of the following is the general solution?

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

Solve the differential equation $y^{\mathrm{\text{'}\text{'}}}+2y^{\mathrm{\text{'}}}=2x+9-e^{-2x}$ using the method of undetermined coefficients. Which of the following is a particular solution?

Which of the following is the general solution to the differential equation $y^{\mathrm{\text{'}\text{'}}}+y=5x\sin \left(x\right)$ using the method of undetermined coefficients?

Explain how a stirred tank reaction works.

Solve the differential equation by separation of variables: $csc\left(y\right) dx+{sec}^2\left(x\right) dy=0$. Which of the following represents the general solution?

Solve the differential equation $y^{\mathrm{\text{'}\text{'}}}+2y=-18x^2{e}^{2x}$ using the method of undetermined coefficients. Which of the following is a particular solution?