13. Intro to Differential Equations

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

${\left(y^{\prime \prime }\right)}^2+6e^t{y}^{\prime }=4t$

Solve the differential equation using variation of parameters: $y^{″}-16y=16x{e}^{4x}$. Which of the following is the general solution?

Solve the differential equation $4y^{″}+y^{\prime }+y=x^2-5x$ using the method of undetermined coefficients. What is the general solution?

Find the general solution of the differential equation $(1+x)\frac{dy}{dx}-xy=x+x^2$.

Of the following, which is not a solution to the differential equation $y^{″}+9y=0$?

Suppose that $y\left(x\right)$ solves the ordinary differential equation $\frac{dy}{dx}=3y$ with the initial condition $y\left(0\right)=2$. What is $y\left(x\right)$?

Find the solution to the differential equation $\frac{dy}{dx}=3y$ with the initial condition $y\left(0\right)=2$.

Solve the differential equation by separation of variables: $\frac{dy}{dx}=\frac{2y+7}{8x+9}$. Which of the following is the general solution?

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

{Use of Tech} Tumor growth The growth of cancer tumors may be modeled by the Gompertz growth equation. Let M(t) be the mass of a tumor, for t ≥ 0. The relevant initial value problem is:

dM/dt = -rM(t)ln(M(t)/K), M(0) = M₀,

where r and K are positive constants and 0 < M₀ < K.

b. Graph the solution for M₀ = 100 and r = 0.05.

Solve the differential equation using variation of parameters: $y^{\text{'}\text{'}}-2y^{\text{'}}+y=e^t\arctan \left(t\right)$. Which of the following is the general solution?

Solve the differential equation: $\frac{dy}{dx}=3x^2{y}^2$. Which of the following is the general solution?

Solve the differential equation $4x^2{y}^{\text{'}\text{'}}+9x{y}^{\text{'}}+y=x^2-x$ using the method of variation of parameters. Which of the following is the general solution?

Find the general solution of the differential equation: $\frac{dr}{d\theta }+r\sec \left(\theta \right)=\cos \left(\theta \right)$.

Solve the differential equation by separation of variables: $\frac{dy}{dx}=x(1-y^2)$. Which of the following is the general solution?

What is the general solution to the differential equation $\frac{dy}{dx}=8e^{4x^2}cos\left(2y\right)$?