Calculus 2

Solve the differential equation $x^2 + 1 \frac{dy}{dx} = x \cdot y$ by separating variables.

Solve a first order differential equation using the method of separation of variables.

Solve the differential equation $y' = \frac{xy}{y^2 + 1}$ using separation of variables.

Solve the differential equation $\frac{dy}{dx} = 3x^2 e^{-y}$ that satisfies the initial condition $y(0) = 0$.

Determine if the infinite series $\sum_{n=1}^{\infty} \frac{5n+3}{7n-4}$ converges or diverges using the divergence test.

Calculate the sum of the first 40 terms of the arithmetic sequence with a first term of 3 and a common difference of 2.

Write a rule for the geometric sequence given two terms: The second term is 6, and the fifth term is 162.

Using the summation notation $\Sigma$, calculate the sum of the geometric series from $k=2$ to $k=7$ with the geometric rule $a_k = \frac{1}{2}^k \times 2$.

Evaluate whether a geometric series with terms A times R^(N-1) is convergent or divergent given different values of R.

Determine if the sequence $a_n = 1 + \frac{(-1)^n}{n^2}$ converges or diverges as $n \to \infty$.

Determine if the sequence $a_n = \sqrt[3]{\frac{n}{n+1}}$ converges or diverges as $n \to \infty$.

Explore whether the infinite series from n equals 1 to infinity of $\frac{1}{n^2}$ converges or diverges using the integral test.

For a series represented with a corresponding function over an interval, use the integral test to determine convergence.

Determine which of the given series might be suitable for an alternating series test, especially those containing terms like $(-1)^n$.

Apply divergence test to a series where the sequence does not converge to zero, such as one involving logarithmic terms.

For series that resemble quotient forms, like $\frac{n}{n^3 + 1}$, determine convergence using the comparison or limit comparison test.

Attempt comparing series with non-standard terms using the limit comparison test.

Utilize the root test for series with terms that include powers like $(\text{something})^n$.

Apply the ratio test to series involving factorial terms and powers, such as those with $n!$ or similar structures.

Use the integral test to determine if the series $\displaystyle \sum_{n=1}^{\infty} \frac{n}{n^2 + 1}$ converges or diverges.