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<p>The variables $w$, $x$ and $y$ are connected by the following differential equations:</p>
<p style="text-align: center">$\frac{\mathrm{d}w}{\mathrm{d}x} = -\frac{3}{2}w - 2 \quad\quad (A)$</p>
<p style="text-align: center">$\frac{\mathrm{d}y}{\mathrm{d}x} = w \qquad\qquad\quad\,\,\, (B)$</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Solve equation (A) to find $w$ in terms of $x$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find $y$ in terms of $x$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Different arbitrary constants found in part <strong>(ii)</strong> yields different solution curves. Some of the solution curves are straight lines. Write down the equation of one of these lines. On a single diagram, sketch your line together with a non-linear solution curve that has your line as an asymptote, indicating clearly any axes intercepts.<span class="inline-marks">[3]</span></div></li></ol>

The variables $w$, $x$ and $y$ are connected by the following differential equations:

$\frac{\mathrm{d}w}{\mathrm{d}x} = -\frac{3}{2}w - 2 \quad\quad (A)$

$\frac{\mathrm{d}y}{\mathrm{d}x} = w \qquad\qquad\quad\,\,\, (B)$

Solve equation (A) to find $w$ in terms of $x$.

Sketching Exponential Functions and Solving Equations Graphically

Solving Differential Equations by Separation of Variables

Solving Differential Equations by Direct Integration

Asymptotes and Discontinuities

Sketching Solution Curves of Differential Equations

Order, General and Particular Solutions of Differential Equations

2016 SAJC P1 Q8 [Modified]

Differential Equations

Graphing Techniques

<p>The variables $x$ and $y$ are related by the differential equation</p>
<p style="text-align: center">$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=2{{\mathrm{e}}^{-2x}}$</p>
<p>Solve the differential equation. Hence find a possible solution curve that has an oblique asymptote, and state the equation of the oblique asymptote.</p><p style="text-align: right;">[4]</p>

The variables $x$ and $y$ are related by the differential equation

$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=2{{\mathrm{e}}^{-2x}}$

Solve the differential equation. Hence find a possible solution curve that has an oblique asymptote, and state the equation of the oblique asymptote.

Second-Order Linear Differential Equations

2023 HCI P1 Q1

<p>Two scientists are investigating the change of a certain population of size $n$ thousand at time $t$ years.</p>
<ol class="i-list"><li><div class="list-item-paragraph">One scientist suggests that $n$ and $t$ are related by the differential equation $\frac{{{\mathrm{d}}^{2}}n}{\mathrm{d}{{t}^{2}}}=10-6t$. Find the general solution of this differential equation. Sketch three members of the family of solution curves, given that $n=100$ when $t=0$.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">The other scientist suggests that $n$ and $t$ are related by the differential equation $\frac{\mathrm{d}n}{\mathrm{d}t}=3-0.02n$. Find $n$ in terms of $t$, given again that $n=100$ when $t=0$. Explain in simple terms what will eventually happen to the population using this model.<span class="inline-marks">[7]</span></div></li></ol>

Two scientists are investigating the change of a certain population of size $n$ thousand at time $t$ years.

Equilibrium Solutions and Long-Term Behaviour

N2009 P2 Q4

<p>The variables $x,y$ and $z$ are connected by the following differential equations.</p>
<p style="text-align: center">$\frac{\mathrm{d}z}{\mathrm{d}x}=3-2z$  $\mathrm{(A)}$</p>
<p style="text-align: center">$\frac{\mathrm{d}y}{\mathrm{d}x}=z$           $\mathrm{(B)}$</p>
<ol class="i-list"><li><div class="list-item-paragraph">Given that $z<\frac{3}{2}$, solve equation $\mathrm{(A)}$ to find $z$ in terms of $x$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Hence find $y$ in terms of $x$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Use the result in part <strong>(ii)</strong> to show that</div><div class="list-item-paragraph" style="text-align: center">$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=a\frac{\mathrm{d}y}{\mathrm{d}x}+b$,</div><div class="list-item-paragraph">for constants $a$ and $b$ to be determined.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph"><strong>* Not in the latest 9758 H2 Math Syllabus</strong></div><div class="list-item-paragraph">The result in part <strong>(ii)</strong> represents a family of curves. Some members of the family are straight lines. Write down the equations of two of these lines. On a single diagram, sketch one of your lines together with a non-linear member of the family of curves that has your line as an asymptote.</div><p style="text-align: right;">[4]</p></li></ol>

The variables $x,y$ and $z$ are connected by the following differential equations.

$\frac{\mathrm{d}z}{\mathrm{d}x}=3-2z$  $\mathrm{(A)}$

$\frac{\mathrm{d}y}{\mathrm{d}x}=z$           $\mathrm{(B)}$

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