Differential Equations (MEI)

Unfortunately a bit late for the exam as it happened last week, but hopefully useful for the next time. A light skim over Differential Equations hopefully able to explain a little if you dont quite understand things. similarly if you dont understand something i've put dont hesitate to leave a comment, as im sure you are not the only one who will get confused. There's no such thing as a stupid question! Enjoy! Always seems to be a lack of MEI material on here

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• Differential Equations (MEI)
• Modelling with differential equations
• The most common way that differential equations will be used to model situations is in Newtons 2nd Law. This is replacing a (acceleration)with dv/dt, v(dv/dx) or (d^2x/dt^2) depending on what you are requiring
• For modelling Simple Harmonic Motion (SHM), the second differential of displacementwill always be used (d^2x/dt^2), and the equation will be of the form d^2x/dt^2+Q^2x=0, where Q is the angular speed of the system. This will always go to x= ACos(Qt)+Bsin(Qt)
• Another example of modelling scenarios with differentials is if there is a resistive force in terms of the velocity or time.
• Tangent Fields
• There will be an insert for this question, and it is a grid of the gradient at each point, represented by a short line. What you are essentially doing is connecting the lines, and drawing the trend that they fit and following the gradients.
• Isoclines are lines where the gradient is a constant value of x or y.
• First Order Differentials
• Seperating Variables
• If dy/dx=f(x)g(y) then this can be rearranged to (1/g(y))dy=f(x)dx, and thus can be integrated. this can only be done if variables can be brought singularly to one side each.
• Integrating Factor
• Used when seperating variables cannot be used. cannot be done if x^2 however. if the equation is of the form dy/dx+f(x)y=g(x), then the integrating factor is e^(int(f(x)dx)),so the equation rearranges to d/dx(e^(int(f(x)dx))y)= g(x)e^int(f(x)).
• Complimentary function and Particular integral can also be used for first orders.
• For an equation of form dy/dx+ay=0, this be rewritten by substituting P for dy/dx, so y=1. So this goes to P+a=0, so P=-a. this means that the solution to this equation is y=Ae^(-ax).
• Second Order Differentials
• Complementary function
• For an equation of the form a(d^2y/dx^2)+bdy/dx+cy=0, this can be rewritten in an auxiliary equation, aP^2+bP+c=0. This then can be solved as a quadratic and the roots of that quadratic become the powers of e. is the roots are R and S, the solution would be y= Ae^(Rx)+Be^(Sx).
• For repeated roots, the soluton takes the form y=(ax+b)e^Rx where R is the repeated Root
• Particular Integral
• Used in conjunction with the complementary fuction, hence the name 'complementary'. When the differential equation doe not =0, i.e. = a function of x.
• When the equation equals a polynomial (Ax^2+bx+c, or even ax+b) then you let y=constants times every power of x of the polynomial. for for =x^3, let y=ax^3+bx^2+cx+d. differentiate this expression as many as is necessary and then substitute into the original differential and equate constants.
• When the equation =trigonometry, (sin or cos, cant do any others) then let y= Asin(x)+bCos(x), and differentiate and substitute again. this is because even if it is just sin or cos, you dont know where they disappear in the differentials. the same with polynomials.
• When the equation =e^power equal to one of the roots, let y=Axe^root. this is because when you differentiate, the product rule means you still have values left after substitution, for the PI will equate to 0 without. When it is a repeated root , then let y=Ax^2e^root.
• Simple Harmonic Motion
• For modelling Simple Harmonic Motion (SHM), the second differential of displacementwill always be used (d^2x/dt^2), and the equation will be of the form d^2x/dt^2+Q^2x=0, where Q is the angular speed of the system. This will always go to x= ACos(Qt)+Bsin(Qt)
• Damping
• Underdamping is where the determinant of the equation is less than 0. e.g. for an auxiliary equation a(P^2)+bP+c=0, if b^2-4ac0