Further Maths - Differential Equations


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Flashcards

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
\[\frac{dy}{dx} = \frac{y + 1}{x}\]

??

  • Seperate the two variables and put them on either side of the equation.
  • Integrate both sides with respect to $x$.
  • Rearrange.

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\[\int \frac{1}{y + 1} \frac{dy}{dx} dx\]

What does this simplify down to??

\[\int \frac{1}{y + 1} dy\]

2021-06-08

What’s the general process for solving a differential equation question like

\[\frac{dy}{dx} = \frac{y + 1}{x}\]

?


  • Seperate the two variables and put them on either side of the equation.
  • Integrate both sides with respect to $x$.
  • Rearrange.

#####

\[\frac{dy}{dx} + P(x)y = Q(x)\]

What’s the formula for the integrating factor??

\[e^{\int P(X) dx}\]
\[\int \frac{1}{y + 1} \frac{dy}{dx} dx\]

What does this simplify down to?


\[\int \frac{1}{y + 1} dy\]

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\[x^2 \frac{dy}{dx} + 2xy = 2x + 1\]

How could you rewrite this??

\[\frac{d}{dx}\left(y \cdot x^2\right) = 2x + 1\]

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\[x^2 e^y \frac{dy}{dx} + 2x e^y = x\]

How could you rewrite this??

\[\frac{d}{dx}\left( x^2 e^y \right)\]

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\[\frac{d}{dx}\left( y^2 \right)\]

What’s this equal to??

\[2y\frac{dy}{dx}\]

2021-06-09

What’s the general form of differential equation where you can use an integrating factor?


\[\frac{dy}{dx} + P(x)y = Q(x)\]

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\[e^{\int 2 dx} = e^{2x + c} = Ae^{2x}\]

What’s gone wrong working of the integrating factor here?? $c$ has been accounted for but it shouldn’t have been.

2021-06-10

\[\frac{dy}{dx} + P(x)y = Q(x)\]

What’s the formula for the integrating factor?


\[e^{\int P(X) dx}\]

What does the integrating factor do?


Turns the left-hand side of a differential equation into the “output” of the product rule.

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\[a \frac{dy}{dx} + by = 0\]

What does the general solution to this look like??

\[Ae^{kx}\]

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\[a\frac{d^2 y}{dx^2} + b\frac{dy}{dx} + cy = 0\]

What does the general solution to this look like??

\[Ae^{\lambda x} + Be^{\mu x}\]

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\[Ae^{\lambda x} + Be^{\mu x}\]

What do $\lambda$ and $\mu$ represent?? Constants to be determined that will solve the differential equation.

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\[y = Ae^{\lambda x} + Be^{\mu x}\]

What is $\frac{dy}{dx}$??

\[\frac{dy}{dx} = A\lambda e^{\lambda x} + B\mu e^{\mu x}\]

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\[y = Ae^{\lambda x} + Be^{\mu x}\]

What is $\frac{d^2y}{dx^2}$??

\[\frac{d^2y}{dx^2} = A\lambda^2 e^{\lambda x} + B\mu^2 e^{\mu x}\]

#####

\[y = Ae^{\lambda x} + Be^{\mu x}\]

If you differentiate and substitute this into

\[a\frac{d^2 y}{dx^2} + b\frac{dy}{dx} + cy = 0\]

What do you get??

\[Ae^{\lambda x} (a\lambda^2 + b\lambda + c) + Be^{\mu x} (a\mu^2 + b\mu + c) = 0\]

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\[Ae^{\lambda x} (a\lambda^2 + b\lambda + c) + Be^{\mu x} (a\mu^2 + b\mu + c) = 0\]

If this is true, what equations can you write for $\lambda$ and $\mu$??

\[a\lambda^2 + b\lambda + c \\\\ a\mu^2 + b\mu + c\]

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\[a\frac{d^2 y}{dx^2} + b\frac{dy}{dx} + cy = 0\]

What is the “auxillary equation”??

\[am^2 + bm + c = 0\]
In the auxillary equation for a differential equation
\[am^2 + bm + c = 0\]

What does $m$ represent?? The values of $\lambda$ or $\mu$.

The auxillary equation for a differential equation is
\[am^2 + bm + c = 0\]

If

\[b^2 - 4ac > 0\]

What does the general solution look like??

\[Ae^{\lambda x} + Be^{\mu x}\]
The auxillary equation for a differential equation is
\[am^2 + bm + c = 0\]

If

\[b^2 - 4ac = 0\]

What does the general solution look like??

\[(A + Bx)e^{\lambda x}\]
The auxillary equation for a differential equation is
\[am^2 + bm + c = 0\]

If

\[b^2 - 4ac < 0\]

What does the general solution look like??

\[Ae^{px}(A\cos qx + B\sin qx)\]

Where the roots are $p \pm q$.

2021-07-07

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\[a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x)\]

Why is this non-homogeneous?? Because it’s equal to a function rather than $0$.

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\[a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x)\]

What two things do you need to add together to solve this for $y$??

  • The complementary function
  • The particular integral
\[x^2 \frac{dy}{dx} + 2xy = 2x + 1\]

How could you rewrite this?


\[\frac{d}{dx}\left(y \cdot x^2\right) = 2x + 1\]
If
\[f(x) = p\]

What is the particular integral??

\[y = \lambda\]
If
\[f(x) = 10\]

What is the form of the particular integral??

\[y = \lambda\]
If
\[f(x) = p + qx\]

What is the particular integral??

\[y = \lambda + \mu x\]
If
\[f(x) = 3x + 2\]

What is the form of the particular integral??

\[y = \lambda + \mu x\]
If
\[f(x) = p + qx + rx^2\]

What is the particular integral??

\[y = \lambda + \mu x + v x^2\]
If
\[y(x) = (x + 2)^2\]

What is the form of the particular integral??

\[y = \lambda + \mu x + v x^2\]
If
\[f(x) = pe^{kx}\]

What is the particular integral??

\[y = \lambda e^{kx}\]
If
\[f(x) = 10e^{-5x)\]

What is the form of the particular integral??

\[y = \lambda e^{-5x}\]
If
\[f(x) = p\cos(\omega x) + q\sin(\omega x)\]

What is the particular integral??

\[y = \lambda \cos (\omega x) + \mu \sin (\omega x)\]
If
\[f(x) = 10\cos\left(\frac{1}{2}x\right)\]

What is the form of the particular integral??

\[y = \lambda \cos\left(\frac{1}{2}x\right) + \mu \sin\left(\frac{1}{2}x\right)\]
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 $\lambda e^{2x}$??

\[y = \lambda x e^{2x}\]
\[x^2 e^y \frac{dy}{dx} + 2x e^y = x\]

How could you rewrite this?


\[\frac{d}{dx}\left( x^2 e^y \right)\]
\[\frac{d}{dx}\left( y^2 \right)\]

What’s this equal to?


\[2y\frac{dy}{dx}\]

2021-09-10

What do you ignore when working out the integrating factor?


The constant of integration, $c$.

2022-01-11

\[e^{\int 2 dx} = e^{2x + c} = Ae^{2x}\]

What’s gone wrong working of the integrating factor here?


$c$ has been accounted for but it shouldn’t have been.

2022-01-19

Given
\[y = e^{px}(A\cos(kx) + B\sin(kx))\]

what is the formula for $\frac{\text{d}y}{\text{d}x}$??

\[\frac{\text{d}y}{\text{d}x} = e^{px}((Ap + Bk)\cos(kx) + (Bp-Ak)\sin(kx))\]
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 $\sin$??

PHOTO 2ND ORDER GRID
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

\[y = (At + B)e^{-t}\]

??

\[y = \lambda t^2 e^{-t} + \mu\]

What does a second-order homogenous differential equation look like?


\[a\frac{d^2 y}{dx^2} + b\frac{dy}{dx} + cy = 0\]

2022-05-15

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?


\[Ae^{kx}\]



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