Further Maths - Differential Equations
Pearson Edexcel Further Mathematics 2022
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Flashcards
Separable first-order equations
What type of answer do you normally get for differential equation questions?
A family of curves.
What’s the general process for solving a differential equation question like
\[\frac{dy}{dx} = \frac{y + 1}{x}\]
?
- Separate the two variables and put them on either side of the equation.
- Integrate both sides with respect to $x$.
- Rearrange.
\[\int \frac{1}{y + 1} \frac{dy}{dx} dx\]
What does this simplify down to?
Integrating factors
What’s the general form of differential equation where you can use an integrating factor?
\[\frac{dy}{dx} + P(x)y = Q(x)\]
What’s the formula for the integrating factor?
What does the integrating factor do?
Turns the left-hand side of a differential equation into the “output” of the product rule.
\[x^2 \frac{dy}{dx} + 2xy = 2x + 1\]
How could you rewrite this?
\[x^2 e^y \frac{dy}{dx} + 2x e^y = x\]
How could you rewrite this?
What do you ignore when working out the integrating factor?
The constant of integration, $c$.
\[e^{\int 2 dx} = e^{2x + c} = Ae^{2x}\]
What’s gone wrong working of the integrating factor here?
$c$ has been accounted for but it shouldn’t have been.
Second-order homogeneous equations
What does a second-order homogenous differential equation look like?
Why do you need two arbitrary constants for the general solution of second-order homogenous differential equations?
Since differentiating twice can remove constant terms and $x$ terms.
\[a \frac{dy}{dx} + by = 0\]
What does the general solution to this look like?
\[a\frac{d^2 y}{dx^2} + b\frac{dy}{dx} + cy = 0\]
What does the general solution to this look like?
\[Ae^{\lambda x} + Be^{\mu x}\]
What do $\lambda$ and $\mu$ represent?
Constants to be determined that will solve the differential equation.
\[y = Ae^{\lambda x} + Be^{\mu x}\]
What is $\frac{dy}{dx}$?
\[y = Ae^{\lambda x} + Be^{\mu x}\]
What is $\frac{d^2y}{dx^2}$?
\[y = Ae^{\lambda x} + Be^{\mu x}\]
If you differentiate and substitute this into
\[a\frac{d^2 y}{dx^2} + b\frac{dy}{dx} + cy = 0\]
What do you get?
\[Ae^{\lambda x} (a\lambda^2 + b\lambda + c) + Be^{\mu x} (a\mu^2 + b\mu + c) = 0\]
If this is true, what equations can you write for $\lambda$ and $\mu$?
\[a\frac{d^2 y}{dx^2} + b\frac{dy}{dx} + cy = 0\]
What is the “auxiliary equation”?
In the auxiliary equation for a differential equation
\[am^2 + bm + c = 0\]
What does $m$ represent?
The values of $\lambda$ or $\mu$.
The auxiliary equation for a differential equation is
\[am^2 + bm + c = 0\]
If
\[b^2 - 4ac > 0\]
What does the general solution look like?
The auxiliary equation for a differential equation is
\[am^2 + bm + c = 0\]
If
\[b^2 - 4ac = 0\]
What does the general solution look like?
The auxiliary equation for a differential equation is
\[am^2 + bm + c = 0\]
If
\[b^2 - 4ac < 0\]
What does the general solution look like?
Where the roots are $p \pm q$.
Non-homogeneous equations and the particular integral
\[a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x)\]
Why is this non-homogeneous?
Because it’s equal to a function rather than $0$.
\[a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x)\]
What two things do you need to add together to solve this for $y$?
- The complementary function
- The particular integral
Why do you need to add together the complementary function and the particular integral for the entire general solution?
The complementary function will always equal zero when put into the equation, so have no bearing on what it equals. The particular integral covers the bit of the function that isn’t $0$.
If
\[f(x) = 10\]
What is the form of the particular integral?
If
\[f(x) = p + qx\]
What is the particular integral?
If
\[f(x) = 3x + 2\]
What is the form of the particular integral?
If
\[f(x) = p + qx + rx^2\]
What is the particular integral?
If
\[y(x) = (x + 2)^2\]
What is the form of the particular integral?
If
\[f(x) = pe^{kx}\]
What is the particular integral?
If
\[f(x) = 10e^{-5x}\]
What is the form of the particular integral?
If
\[f(x) = p\cos(\omega x) + q\sin(\omega x)\]
What is the particular integral?
If
\[f(x) = 10\cos\left(\frac{1}{2}x\right)\]
What is the form of the particular integral?
For the differential equation
\[\frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 6y = e^{2x}\]
The complementary function has been found to be
\[Ae^{2x} + Be^{2x}\]
What should use instead of $\lambda e^{2x}$?
When might you have to modify the normal templates for the particular integral when working out a differential equation?
If the complementary function shares some “element” of the particular integral.
If you can’t use $y = \lambda$ as the particular integral since it shares some “element” of $f(x)$ in a differential equation, what would you use instead?
Modelling and related rates
If $x$ is the amount of copper sulphate in a tank, how could you think about $\frac{\text{d}x}{\text{d}t}$?
If $v = e^{\frac{x}{2}}$ and you wanted to find $x$ as a function of $t$, how could you approach this?
Flip the derivative.
\[\frac{\text{d}x}{\text{d}t} = e^{-\frac{x}{2}}\]to
\[\frac{\text{d}t}{\text{d}x} = e^{\frac{x}{2}}\]Derivative of the oscillatory general solution
Given
\[y = e^{px}(A\cos(kx) + B\sin(kx))\]
what is the formula for $\frac{\text{d}y}{\text{d}x}$?
Given
\[y = e^{px}(A\cos(kx) + B\sin(kx))\]
what is the nice “grid” for remembering the coefficients of the derivative for $\cos$ and $\sin$?
What would the format of the particular integral be for
\[\frac{\text{d}^2y}{\text{d}t^2} + 2 \frac{\text{d}y}{\text{d}t} + y = e^{-t} + 1\]
if the complementary function is
\[y = (At + B)e^{-t}\]
?
When powering up a particular integral, do you power up everything or only the bits that might interact?
Only the bits that might interact.
The derivative grid mnemonic
What’s a stupid mnemonic to remember the grid to quickly work out the derivative of a general solution to a second order differential equation?
A Basketball Player, Killed by an AK
Using the stupid mnemonic “a basketball player, killed by an AK”, can you visualise the grid for finding the derivative of the general solution to a second order differential equation?