# Recent questions in Differential equations

Differential equations

### Transform the second-order differential equation $$\displaystyle{\frac{{{d}^{{{2}}}{x}}}{{{d}{t}^{{{2}}}}}}={3}{x}$$ into a system of first-order differential equations.

Differential equations

### True or False? Justify your answer with a proof or a counterexample. You can determine the behavior of all first-order differential equations using directional fields or Euler's method.

Differential equations

### Use the substitution $$y'=v$$ to write each second-order equation as a system of two first-order differential equations (planar system). $$\displaystyle{y}^{''}+\mu{\left({t}^{{{2}}}-{1}\right)}{y}^{''}+{y}={0}$$

Differential equations

### Suppose that a population develops according to the logistic equation $$\frac{dP}{dt}=0.05P-0.0005P^2$$ where t is measured in weeks. What is the carrying capacity? What is the value of k?

Differential equations

### If $$x^2 + xy + y^3 = 1$$ find the value of y''' at the point where x = 1

Differential equations

### Find $$\frac{dy}{dx}$$ using implicit differentiation $$xe^{y}=x-y$$

Differential equations

### Consider the differential equation for a function f(t), $$tf"(t)+f'(t)-f((t))^2=0$$ a) What is the order of this differential equation? b) Show that $$f(t)=\frac{1}{t}$$ is a particular solution to this differential equation. c) Find a particular solution with $$f(0)=0$$ 2. Find the particular solutions to the differential equations with initial conditions: a)$$\frac{dy}{dx}=\frac{\ln(x)}{y}$$ with $$y(1)=2$$ b)$$\frac{dy}{dx}=e^{4x-y}$$ with $$y(0)=0$$

Differential equations

### Transform the second-order differential equation $$\displaystyle{\frac{{{d}^{{{2}}}{x}}}{{{d}{t}^{{{2}}}}}}+{\frac{{{d}{x}}}{{{d}{t}}}}={x}$$ into a system of first-order differential equations.

Differential equations

### Transform the single linear differential equation into a system of first-order differential equations. $$\displaystyle{x}{'''}+{t}{x}{''}+{2}{t}^{{{3}}}{x}'-{5}{t}^{{{4}}}={0}$$

Differential equations

### $$\displaystyle{y}^{''}+{3}{y}^{''}+{4}{y}={2}{\cos{{2}}}{t}$$

Differential equations

### True or False? Justify your answer with a proof or a counterexample. You can explicitly solve all first-order differential equations by separation or by the method of integrating factors.

Differential equations

### Show that the second-order differential equation $$y″ = F(x, y, y′)$$ can be reduced to a system of two first-order differential equations $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={z},{\frac{{{\left.{d}{z}\right.}}}{{{\left.{d}{x}\right.}}}}={F}{\left({x},{y},{z}\right)}.$$ Can something similar be done to the nth-order differential equation $$\displaystyle{y}^{{{\left({n}\right)}}}={F}{\left({x},{y},{y}',{y}{''},\ldots,{y}^{{{\left({n}-{1}\right)}}}\right)}?$$

Differential equations

### if tax revenue is $230 billion and the government's outlays are$235 billion, then the budget a) surplus is $230 billion and the budget deficit is$235 billion. b) deficit is $5 billion and government debt will remain the same. c) surplus is$5 billion and government debt will increase by $5 billion. d) surplus is$230 billion and government debt will decrease by 5$billion. e) deficit is 5$ billion and government debt will increase by 5\$ billion.

Differential equations

### If $$xy+6e^y=6e$$ , find the value of y" at the point where x=0

Differential equations

### Use the substitution $$v=y'$$ to write each second-order equation as a system of two first-order differential equations (planar system). $$y''+2y'-3y=0$$

Differential equations

### Solve the given differential equation by separation of variables: $$\frac{dy}{dx}=x\sqrt{1-y^2}$$

Differential equations

### Use the substitution $$v=y'$$ to write each second-order equation as a system of two first-order differential equations (planar system). $$4y''+4y'+y=0$$

Differential equations

### Solve the differential equation by variation of parameters $$y" + 3y' +2y = \frac{1}{1+e^x}$$

Differential equations