9709 · 3.8
Differential equations flashcards
Revision flashcards for Cambridge 9709 Differential equations (syllabus 3.8). Flip, recall, then mark a real past-paper question.
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What is a differential equation?
An equation that contains a derivative, such as $dy/dx$, linking a function with one or more of its derivatives.
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What is the 'order' of a differential equation?
The order of the highest derivative present in the equation. A-Level Maths P3 focuses on first-order equations, which involve $dy/dx$.
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What is a 'general solution' to a differential equation?
A solution that includes an arbitrary constant of integration, usually denoted by $c$. It represents a family of curves.
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What is a 'particular solution'?
A unique solution obtained by using a given boundary condition (e.g., $y=2$ when $x=1$) to find the specific value of the constant of integration $c$.
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When can you use the 'separation of variables' method?
When the differential equation can be written in the form $dy/dx = f(x)g(y)$. You can then rearrange it to have all $y$ terms on one side and all $x$ terms on the other.
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What is the most common mistake when separating variables?
Forgetting to add the constant of integration, $+\,c$, immediately after performing the integration on one side (usually the x-side).
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What is the standard form for using an integrating factor?
The linear first-order form: $dy/dx + P(x)y = Q(x)$. The coefficient of $dy/dx$ must be 1.
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What is the formula for the integrating factor (I.F.)?
The integrating factor is $\text{I.F.} = e^{\int P(x) dx}$. Note that you don't need a constant of integration when finding the I.F.
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After multiplying by the I.F., what does the left side of $dy/dx + P(x)y = Q(x)$ become?
It becomes the derivative of the product of $y$ and the integrating factor: $d/dx (y \cdot \text{I.F.})$.
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What form does a statement like "the rate of increase of N is proportional to N" take?
This translates to the differential equation $dN/dt = kN$, where $k$ is a positive constant of proportionality.
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Why is $\ln|y|$ important when integrating $1/y$?
The integral of $1/y$ is $\ln|y|$. The modulus is crucial because the argument of a logarithm must be positive. You often resolve the modulus using the initial conditions.