Y= 58x + 55
They charge 58$ per hour therefore 58 should correlate with x and since when don’t know y (the total amount) its y= and then just add 55
A short-cut to accurately evaluate the given expression above is using a scientific calculator where one can include integrals and evaluate using limits. In this case, using a calculator, the answer is equal to 0.2679. One can verify this by integrating truly letting 1-x^2 as u and use its du to be substituted in the numerator
The given differential equation is
<em>x</em> ² (<em>y'</em> - <em>x</em> ²) + 3<em>xy</em> = cos(<em>x</em>)
Expanding and rearranging terms, we get
<em>x</em> ² <em>y'</em> + 3<em>xy</em> = cos(<em>x</em>) + <em>x</em> ⁴
Multiply both sides by <em>x</em>, which is motivated by the fact that (<em>x</em> ³)<em>'</em> = 3<em>x</em> ².
<em>x</em> ³ <em>y'</em> + 3<em>x</em> ²<em>y</em> = <em>x</em> cos(<em>x</em>) + <em>x</em> ⁵
The left side is the derivative of a product:
(<em>x</em> ³<em>y</em>)<em>'</em> = <em>x</em> cos(<em>x</em>) + <em>x</em> ⁵
Integrate both sides with respect to <em>x</em> :
∫ (<em>x</em> ³<em>y</em>)<em>'</em> d<em>x</em> = ∫ (<em>x</em> cos(<em>x</em>) + <em>x</em> ⁵) d<em>x</em>
<em>x</em> ³<em>y</em> = cos(<em>x</em>) + <em>x</em> sin(<em>x</em>) + 1/6 <em>x</em> ⁶ + <em>C</em>
Solve for <em>y</em>. Since <em>x</em> > 0, we can safely divide both sides by <em>x</em> ³.
<em>y</em> = cos(<em>x</em>)/<em>x</em> ³ + sin(<em>x</em>)/<em>x </em>² + 1/6 <em>x</em> ³ + <em>C</em>/<em>x</em>³
Point a has the ordered pair of (3,5)
