13. Intro to Differential Equations

Which of the following is the general solution to the differential equation $3x^2{y}^{\text{'}\text{'}}+6x{y}^{\text{'}}+y=0$ for $x>0$?

Solve the differential equation $e^{x}y\frac{dy}{dx}=e^{-y}+e^{-4x}-y$ by separation of variables.

Solve the differential equation $3x^2{y}^{″}+7x{y}^{\prime }+y=x^2-x$ using the method of variation of parameters. Which of the following is the general solution?

Newton’s Law of Cooling A cup of coffee is removed from a microwave oven with a temperature of 80°C and allowed to cool in a room with a temperature of 25°C. Five minutes later, the temperature of the coffee is 60°C.

c. When does the temperature of the coffee reach 50°C?

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

a. The general solution of the differential equation y'(t) = 1 is y(t) = t

Find the general solution of the system of differential equations:

$$\frac{\partial x}{\partial t}=-5x+4y$$$$\frac{\partial y}{\partial t}=3x-3y$$

Solve the differential equation using variation of parameters: $y^{″}+3y^{\prime }+2y=15+e^x$. Which of the following is the general solution?

Solve the differential equation $x^2{y}^{\text{'}\text{'}}-3x{y}^{\text{'}}+3y=x^4{e}^x$ using the method of variation of parameters. Which of the following is the general solution?

Find the general solution of the differential equation: $x^2{y}^{\prime }+x(x+2)y=e^x$.

Find the general solution of the differential equation $y^4+y^3+y^2=0$.

Solve the differential equation by variation of parameters: $y^{″}+y=\sin ⁠2x$. What is the general solution $y\left(x\right)$?

Find the general solution of the differential equation: ${\cos }^2\left(x\right)\sin \left(x\right)\frac{dy}{dx}+({\cos }^3(x\left)\right)y=1$.

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?