IZTECH Math255 Notes p3

By E. Z.

Constant Coefficient Homogenous Eq.

\(\ a,b ∈ ℝ,a ≠ 0\)

\(\ ay''+by'+cy=0 ⇒ y''+\frac{b}{a}y'+\frac{c}{a}y=0 \)

\(\ p(x)=\frac{b}{a} , q(x)=\frac{c}{a} \)

So that 𝓅 and 𝓆 are continous functions on real line, then there exists a unique solution to the IVP: \(\ ay''+by'+cy=0, y(x_0)=y_0, y'(x_0)=y_1 \)
\(\ y'',y',y \) are linearly dependent for DE: \(\ ay''+by'+cy=0\)
\(\ y=e^{rx} ⇒ e^{rx}[ar^2+br+c]=0 \)
For \(\ ar^2+br+c=0\) there is 3 options:
  • Real and distinct roots
  • Real and repeated roots
  • Complex conjugate roots

    • Real and distinct roots

    \(\ y_1=e^{r_1x},y_2=e^{r_2x} ⇒ y=c_1e^{r_1x}+c_2e^{r_2x} \)

    \(\ y_1 \) and \(\ y_2 \) are linearly independent

    Real and repeated roots

    \(\ r_1=r_2=- \frac{b}{2a} ⇒ y''-2ry+r^2y=0 ⇒ y=c_1e^{rx}+c_2xe^{rx} \)

    Complex conjugate roots

    \(\ r_1=α+iβ,r_2=α-iβ ⇒ y_1=e^{αx}cos(βx), y_2=e^{αx}sin(βx) \) then:
    \(\ y=c_1e^{αx}cos(βx)+c_2e^{αx}sin(βx) \)

    Complex Conjugate Roots for Characteristic Eq.

  • α > 0 → no limit with growning magnitude
  • α = 0 → pure oscillation
  • α < 0 → magnitude of oscillation vanishes


    • as x → ∞

    α > 0

    α = 0

    α < 0


    Mechanical Vibrations

    Differential Model

    \(\ \vec{F}_{net}=\vec{F}_{gravity}+\vec{F}_{spring}+\vec{F}_{damping} \)

    \(\ \vec{F}_{gravity}=m\vec{g}, \vec{F}_{spring}=-k \vec{x}(t), \vec{F}_{damping}=-b \vec{v}(t), \vec{F}_{net}=m \vec{a}(t) \)

    \(\\ 0=mu''+ bu' + ku \)

    where m is mass, k is stiffness of spring and b is damping coefficient

    if b = 0 it is called to be damped
    if b ≠ 0 it is called to be undamped
    for:
    \(\\ \small 0=mu''+ bu' + ku \)

    m is proportional to periodity of motion
    k is inversely proportional to periodity of motion

    From characteristic eq.: \(\ r_{1,2}= \frac{-b \pm \sqrt{b^2-4km}}{2m} \)

    if \(\ b^2-4km \) > 0 → overdamped oscillation

    if \(\ b^2-4km \) = 0 → critically damped oscillation

    if \(\ b^2-4km \) > 0 → underdamped oscillation

    Constant Coefficient Non-homogenous Eq.

    \(\ ay''+by'+cy=f(x) \)
    \(\ p(x), q(x), f(x) \) are continous functions on open interval 𝕀 & \(\ \small x_0 ∈ \) 𝕀. Given two values \(\ \small y_0 \text{ and } y_1\), there exists a unique solution to the IVP on the open interval 𝕀 :

    \(\ y''+p(x)y'+q(x)y=f(x), y(x_0)=y_a, y'(x_0)=y_b \)
    For homogenous part (f(x)=0) has solution in the form: \(\ c_1 y_1 + c_2 y_2=y_h=y_c \)
    The particular solution \(\ \small y_p(x) \) is linearly independent to the functions \(\ y_1 \) & \(\ y_2 \).
    0+f(x)=f(x) ⇒ \(\ y_g=y_p+y_h \)

    Method of Undetermined Coefficients

    This method is based on initial guess of dependent variable based on structer of the DE.

    \(\ y''-3y'+2y=2e^{3x} \) supposing \(\ y(x)=Ae^{3x} \), A=1 so y= \(\ e^{3x} \)
    \(\ y''+4y'+5y=x^2-1 \)
    \(\ r_{1,2}=-2 ±i ⇒ α=-2 , β=1 \)
    \(\ y_h=e^{-2x}(c_1 cos (x)+c_2sin(x)) \)
    \(\ y_p=Ax^2+Bx+C \) from putting it into DE:
    \(\ (5A)x^2 +(8A+5B)x + (2A+4B+5C)=x^2-1 \)
    \(\ \begin{bmatrix} 5, 0, 0 \\8, 5, 0 \\2, 4, 5 \end{bmatrix} \begin{bmatrix} A\\B\\C \end{bmatrix}= \begin{bmatrix} 1\\ 0\\-1 \end{bmatrix} \)
    \(\ y_g=y_p+y_h \)
    f(x) x
    Exponential Exponential with coefficient and same x coefficient
    Polinomial Polinomial of the same degree with unknown coefficients
    Trigonometric Linear Combinations of trigonometric functions with unknown coefficients


    for such 𝒻 containing their multiplications, 𝓍 is multiplication of them.

    Variation of Parameters

    It is a solution method for non-homogenous DE's. It is harder to apply than Undetermined Coefficients Method, but it does not requires any assumption and works for greater range of DE's including ones can be solved using undetermined coefficients method.

    \(\ y'' + 4y =3cosec(t) \)


    It is hard to guess a particular solution for a t using unknown coefficients method.

    Strategy
    for DE: \(\ ay''+by'+cy=f(x) \);
    \(\ \small y_h=c_1y_1(x)+c_2y_2(x) \) and \(\ \small y_p=u_1(x)y_1(x)+u_2(x)y_2(x) \)

    \(\ \small u_1= ∫ - \frac{f(x).y_2}{a.W(y_1,y_2)} dx \) and \(\ \small u_2= ∫ \frac{f(x).y_1}{a.W(y_1,y_2)} dx \)

    Proof is left as an exercise for the reader.

    \(\ y''+16y=sec(4x) \)


    characteristic equation is: \(\ r^2+16=0, α=0, β=4 \)
    \(\ y_h=c_1cos(4x)+c_2sin(4x) \), \(\ y_p= \frac{cos(4x)ln(cos(4x))}{16}+\frac{sin(4x)x}{4} \)


    \(\ y_g=c_1cos(4x)+c_2sin(4x)+\frac{cos(4x)ln(cos(4x))}{16}+\frac{sin(4x)x}{4} \)