Differential Equations

Apply integration by parts on either side before attempting separation of variables.

Integrate both sides directly then apply initial conditions secondarily.

Separate variables before integrating each side independently.

Use partial fraction decomposition on either integral prior to addressing variables separately.

Increasing the value of $L$

Increasing both values of $k$ and $L$

Decreasing the value of $k$

Decreasing both values of $k$ and leaving $L$ constant

$y = \frac{-1}{\cos(x) + C}$, where $C$ is a constant.

$y = \frac{1}{x - C}$, where $C$ is a constant.

$y = \frac{-1}{x + C}$, where $C$ is a constant.

$y = \frac{-1}{x^2 + C}$, where $C$ is a constant.

Integrate the expression $\int{\frac{dr}{r^2}}$ and $\int{\cos\theta}d\theta$ to find the general solution.

Set up the definite integrals with limits from $0$ to $1$ for $r(\theta)$ and from $0$ to $\theta$ for $\theta$ respectively before integration.

Use partial fractions decomposition on the left-hand side expression before integration.

Invert the relationship to get $\int{\frac{da}{ra}} = \int{\sin\alpha}d\alpha$ before integrating.

$\sin(5)$

$\frac{1}{5}$

$1$

$0$

To satisfy mathematical constraints

To account for physical constraints

To obtain a unique solution

To simplify the equation

y = 4e^x + 1

y = 2e^x + 1

y = 2e^x

y = 4e^x

$f'(a)$ does not exist.

$f(x)$ is continuous at $x = a$.

There is a removable discontinuity at $x = a$.

The left-hand limit of $f(x)$ as x approaches 'a' does not equal the right-hand limit.

(pi - square root three plus one)

(-pi plus square root pi four over Three)

(-pi - square root three plus one)

(-Pi Minus Pi four over three)

Undefined solution

General solution

Infinite solution

Particular solution