Calculus Examples

$\frac{d T}{d t} + \frac{kT}{r^{2}} = 0$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Solve for $\frac{d T}{d t}$ .

Tap for more steps...

Step 1.1.1

Subtract $\frac{kT}{r^{2}}$ from both sides of the equation.

$\frac{d T}{d t} = - \frac{kT}{r^{2}}$

Step 1.1.2

Rewrite the expression.

$\frac{d T}{d t} = - \frac{1}{r^{2}} kT$

Step 1.2

Rewrite the equation.

$d T = - \frac{1}{r^{2}} kT d t$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d T = \int - \frac{1}{r^{2}} kT d t$

Step 2.2

Apply the constant rule.

$T + C_{1} = \int - \frac{1}{r^{2}} kT d t$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Apply the constant rule.

$T + C_{1} = - \frac{1}{r^{2}} kT t + C_{2}$

Step 2.3.2

Combine $t$ and $\frac{1}{r^{2}}$ .

$T + C_{1} = - \frac{t}{r^{2}} kT + C_{2}$

Step 2.3.3

Reorder terms.

$T + C_{1} = - kT \frac{t}{r^{2}} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$T = - kT \frac{t}{r^{2}} + K$