This is a linear differential equation of first order. Solve this by integrating the coefficient of the y term and then raising e to the integrated coefficient to find the integrating factor, i.e. the integrating factor for this problem is e^(6x).
<span>Multiplying both sides of the equation by the integrating factor: </span>
<span>(y')e^(6x) + 6ye^(6x) = e^(12x) </span>
<span>The left side is the derivative of ye^(6x), hence </span>
<span>d/dx[ye^(6x)] = e^(12x) </span>
<span>Integrating </span>
<span>ye^(6x) = (1/12)e^(12x) + c where c is a constant </span>
<span>y = (1/12)e^(6x) + ce^(-6x) </span>
<span>Use the initial condition y(0)=-8 to find c: </span>
<span>-8 = (1/12) + c </span>
<span>c=-97/12 </span>
<span>Hence </span>
<span>y = (1/12)e^(6x) - (97/12)e^(-6x)</span>
[True]
Natural numbers are the positive integers (whole numbers) such as 1, 2, 3, and etc. So it is true that zero is less than every natural number.
[Hundreds]
The 8 in 30,846 is in the hundreds place because the 8 stands for 800.
First, plug in 5 for x:
| -5 * 5 + 12 | = |-25 + 12| = |-13|
Those two lines are absolute value, which measures the distance from the number to 0 on a number line. This distance is always positive.
The distance from -13 to 0 on a number line is 13. The answer is 13.
