# Recent questions in Calculus 2

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

Series

Series

### Determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. The function are real-valued. All rational functions.

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

Series

### Which of the following statements are​ true? i.If $$a_{n}\ \text{and}\ f(n)\ \text{satisfy the requirements of the integral test, then} \sum_{n=1}^{\infty} a_n = \int_1^{\infty} f(x)dx$$. ii. The series $$\sum_{n=1}^{\infty} \frac{1}{n^p}\ \text{converges if}\ p > 1\ \text{and diverges if}\ p \leq 1$$. iii. The integral test does not apply to divergent sequences.

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

### Find the exact length of the curve. Use a graph to determine the parameter interval. $$r=\cos^2(\frac{\theta}{2})$$

Differential equations

### Find an equation of the plane. The plane through the points (2,1,2), (3,-8,6), and (-2,-3,1)

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