<p>The variables \(x\) and \(y\) are related by the differential equation</p>
<p style="text-align: center">\[x^2 \frac{\mathrm{d}y}{\mathrm{d}x} - 2xy = 10x^2y - y^2 \quad \text{for}\hspace{0.5em} x &gt; 0.\]</p>
<ol class="i-list"><li><div class="list-item-paragraph">Using the substitution \(y = ux^2\), for&nbsp;\(x &gt; 0\), show that the differential equation can be reduced to</div><div class="list-item-paragraph" style="text-align: center">\[\frac{\mathrm{d}u}{\mathrm{d}x} = u(10 - u).\]<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence show that the general solution for the differential equation</div><div class="list-item-paragraph" style="text-align: center">\[x^2 \frac{\mathrm{d}y}{\mathrm{d}x} - 2xy = 10x^2y - y^2 \quad \text{for}\hspace{0.5em} x &gt; 0\]</div><div class="list-item-paragraph">can be expressed in the form \(y = \frac{10x^2}{Qe^{-10x} + 1}\), where&nbsp;\(Q\) is an arbitrary constant.<span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">Given that the graph of \(y = \mathrm{f}(x)\) passes through the point with coordinates \(\left(\frac{1}{10}, \frac{1}{10(1 - \mathrm{e}^{-1})}\right)\), find the equation of the graph of \(y = \mathrm{f}(x)\).<span class="inline-marks">[2]</span></div></li></ol>

The variables \(x\) and \(y\) are related by the differential equation

\[x^2 \frac{\mathrm{d}y}{\mathrm{d}x} - 2xy = 10x^2y - y^2 \quad \text{for } x > 0.\]

Using the substitution \(y = ux^2\), for \(x > 0\), show that the differential equation can be reduced to

\[\frac{\mathrm{d}u}{\mathrm{d}x} = u(10 - u).\]

Hence show that the general solution for the differential equation

\[x^2 \frac{\mathrm{d}y}{\mathrm{d}x} - 2xy = 10x^2y - y^2 \quad \text{for } x > 0\]

can be expressed in the form \(y = \frac{10x^2}{Qe^{-10x} + 1}\), where \(Q\) is an arbitrary constant.

Given that the graph of \(y = \mathrm{f}(x)\) passes through the point with coordinates \(\left(\frac{1}{10}, \frac{1}{10(1 - \mathrm{e}^{-1})}\right)\), find the equation of the graph of \(y = \mathrm{f}(x)\).

2024 MI PRU2 P2 Q5

2024 MI PRU2 P2 Q5