Given that $$\frac{dy}{dx} = -x^3 + \frac{4x-5}{2x^3}, \quad x \neq 0$$, 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 10 - 2013 - Paper 3

Integrate both sides:

$y = \int \left(-x^3 + \frac{2}{x^2} - \frac{5}{2x^3}\right) \, dx$

This results in:

$y = -\frac{x^4}{4} + 2\int x^{-2} \, dx - \frac{5}{2}\int x^{-3} \, dx$

Perform the integrations:

1. $\int x^{-2} \, dx = -\frac{1}{x}$
2. $\int x^{-3} \, dx = -\frac{1}{2x^2}$

Substituting these back, we have:

$y = -\frac{x^4}{4} - 2\frac{1}{x} + \frac{5}{4x^2} + C$

Using the initial condition $y = 7$ at $x = 1$:

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

This simplifies to:

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

Combining terms gives:

$7 = \frac{6}{4} + C$

So, $C = 7 - \frac{6}{4} = 7 - \frac{3}{2} = \frac{14 - 3}{2} = \frac{11}{2}$

Finally, substituting back $C$ into the expression for $y$, we have:

$y = -\frac{x^4}{4} - \frac{2}{x} + \frac{5}{4x^2} + \frac{11}{2}$

which is the required expression for $y$ in terms of $x$, with each term in its simplest form.