Given that y = 7 at x = 1, find y in terms of x, giving each term in its simplest form - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 3

The integral can be computed as follows:

$y = -\frac{1}{4}x^4 + 2\ln|x| + \frac{5}{2} \cdot \frac{1}{x} + c$

Thus,

$y = -\frac{1}{4}x^4 + 2\ln|x| + \frac{5}{2}x^{-1} + c$

Now we apply the initial condition:

y = 7 at x = 1.
Substituting these values into the equation gives:

$7 = -\frac{1}{4}(1)^4 + 2\ln(1) + \frac{5}{2}(1) + c$

Since ln(1) = 0,
we simplify to:

$7 = -\frac{1}{4} + \frac{5}{2} + c$

This simplifies to
$7 = -\frac{1}{4} + \frac{10}{4} + c$

So,

$7 = \frac{9}{4} + c$

Solving for c yields:

$c = 7 - \frac{9}{4} = \frac{28}{4} - \frac{9}{4} = \frac{19}{4}$

Substituting c back into the equation, we have:

$y = -\frac{1}{4}x^4 + 2\ln|x| + \frac{5}{2}x^{-1} + \frac{19}{4}$

This is the required expression for y in terms of x.