# Recent questions in Differential equations

First order differential equations

### Write first-order differential equations that express the situation: The increase in the number of people who have heard a rumor in a town of population 20,00020,000 is proportional to the product of the number PP who have heard the rumor and the number who have not heard the rumor at time tt .

Laplace transform

### Inverse Laplace transformation $$(s^2 + s)/(s^2 +1)(s^2 + 2s + 2)$$

First order differential equations

### Write first-order differential equations that express the situation: A student in an engineering course finds the rate of increase of his grade point average GG is directly proportional to the number NN of study hours/week and inversely proportional to the amount AA of time spent on online games .

Laplace transform

### Use the table of Laplace transform and properties to obtain the Laplace transform of the following functions. Specify which transform pair or property is used and write in the simplest form. a) $$x(t)=\cos(3t)$$ b)$$y(t)=t \cos(3t)$$ c) $$z(t)=e^{-2t}\left[t \cos (3t)\right]$$ d) $$x(t)=3 \cos(2t)+5 \sin(8t)$$ e) $$y(t)=t^3+3t^2$$ f) $$z(t)=t^4e^{-2t}$$

First order differential equations

### Identify the surface whose equation is given. $$\rho= \sin \theta \sin \phi$$

Second order linear equations

### The integrating factor method, which was an effective method for solving first-order differential equations, is not a viable approach for solving second-order equstions. To see what happens, even for the simplest equation, consider the differential equation $$\displaystyle{y}{''}+{3}{y}'+{2}{y}={f{{\left({t}\right)}}}$$. Lagrange sought a function $$\displaystyle\mu{\left({t}\right)}μ{\left({t}\right)}$$ such that if one multiplied the left-hand side of $$\displaystyle{y}{''}+{3}{y}'+{2}{y}={f{{\left({t}\right)}}}$$ bu $$\displaystyle\mu{\left({t}\right)}μ{\left({t}\right)}$$, one would get $$\displaystyle\mu{\left({t}\right)}{\left[{y}{''}+{y}'+{y}\right]}={d}{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left[\mu{\left({t}\right)}{y}+{g{{\left({t}\right)}}}{y}\right]}$$ where g(t)g(t) is to be determined. In this way, the given differential equation would be converted to $$\displaystyle{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left[\mu{\left({t}\right)}{y}'+{g{{\left({t}\right)}}}{y}\right]}=\mu{\left({t}\right)}{f{{\left({t}\right)}}}$$, which could be integrated, giving the first-order equation $$\displaystyle\mu{\left({t}\right)}{y}'+{g{{\left({t}\right)}}}{y}=\int\mu{\left({t}\right)}{f{{\left({t}\right)}}}{\left.{d}{t}\right.}+{c}$$ which could be solved by first-order methods. (a) Differentate the right-hand side of $$\displaystyle\mu{\left({t}\right)}{\left[{y}{''}+{y}'+{y}\right]}={d}{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left[\mu{\left({t}\right)}{y}+{g{{\left({t}\right)}}}{y}\right]}$$ and set the coefficients of y,y' and y'' equal to each other to find g(t). (b) Show that the integrating factor $$\displaystyle\mu{\left({t}\right)}μ{\left({t}\right)}$$ satisfies the second-order homogeneous equation $$\displaystyle\mu{''}-\mu'+\mu={0}$$ called the adjoint equation of $$\displaystyle{y}{''}+{3}{y}'+{2}{y}={f{{\left({t}\right)}}}$$. In other words, althought it is possible to find an "integrating factor" for second-order differential equations, to find it one must solve a new second-order equation for the integrating factor μ, which might be every bit as hard as the original equation. (c) Show that the adjoint equation of the general second-order linear equation $$\displaystyle{y}{''}+{p}{\left({t}\right)}{y}'+{q}{\left({t}\right)}{y}={f{{\left({t}\right)}}}$$ is the homogeneous equation $$\displaystyle\mu{''}-{p}{\left({t}\right)}\mu'+{\left[{q}{\left({t}\right)}-{p}'{\left({t}\right)}\right]}\mu={0}$$.

First order differential equations

### Write first-order differential equations that express the situation: A Walking Mode, l Construct a mathematical model to estimate how long it will take you to walk to the store for groceries. Suppose you have measured the distance as 11 mile and you estimate your walking speed as 33 miles per hour. If it takes you 2020 minutes to get to the store, what do you conclude about your model?

First order differential equations

### Write the following first-order differential equations in standard form. $$\displaystyle{y}'={x}^{{{3}}}{y}+{\sin{{x}}}$$

First order differential equations

### Find the differential of each function. (a) $$y = \tan \sqrt{t}$$ (b) $$y= \frac{1-v^2}{1+v^2}$$

First order differential equations

### Write the following first-order differential equations in standard form. $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}={y}{x}{\left({x}+{1}\right)}$$

Second order linear equations

### Use the family in Problem 1 to ﬁnd a solution of $$y+y''=0$$ that satisﬁes the boundary conditions $$y(0)=0,y(1)=1.$$

Laplace transform

### Explain why the function is discontinuous at the given number a. Sketch the graph of the function. $$f(x) = \left\{\frac{1}{x+2}\right\}$$ if $$x \neq -2$$ $$a= -2$$ 1 if $$x = -2$$

Laplace transform

### Use the Laplace transform to solve the given initial-value problem. $$dy/dt-y=z,\ y(0)=0$$

Laplace transform

### Find the Laplace transform of the function $$L\left\{f^{(9)}(t)\right\}$$

Second order linear equations

### Solve the equation: $$\displaystyle{\left({a}-{x}\right)}{\left.{d}{y}\right.}+{\left({a}+{y}\right)}{\left.{d}{x}\right.}={0}$$

First order differential equations

### If $$f(x) + x^2[f(x)]^5 = 34$$ and $$f(1) = 2,$$ find $$f '(1).$$

First order differential equations

### How do you solve linear first-order differential equations?

Laplace transform

### Use Theorem 7.4.3 to find the Laplace transform F(s) of the given periodic function. F(s)=?

First order differential equations