Differential equations - Second order ordinary differential equations

  1. Homogeneous Linear Equations with Constant Coefficients 
    \(y''+py'+qy=0.\)
    The characteristic equation is 
    \(\lambda^2+p\lambda+q=0.\)
    If \(\lambda_1\) and \(\lambda_2\) are distinct real roots of the characteristic equation , then the general solution is 
    \(y=C_1e^{\lambda_1x}+C_2e^{\lambda_2x},\) Where 
    \(C_1\) and \(C_2\) are integration contstants.
    If \(\lambda_1=\lambda_2=-{p \over 2},\)then the general solution is 
    \(y=(C_1+C_2x)e^{{-p \over 2}x}.\)
    If \(\lambda_1\) and \(\lambda_2\) are complex numbers :
    \(\lambda_1=\alpha+\beta i,\lambda_2=\alpha- \beta i,\) where
    \(\alpha= -{p \over 2}, \beta= {\sqrt {4q-p^2} \over 2},\)
    then the general solution is 
    \(y=e^{\alpha x}(C_1cos\beta x+C_2 sin \beta x).\)
  2. Inhomogeneous Lineaar Equations with Constant Coefficints.
    \(y''+py'+qy=f(x).\)
    The general solution is given by
    \(y=y_p+y_h,\) where
    \(y_p\) is a particular solution of the inhomogeneous equation and \(y_h\) is the general solution of the associated homogeneous equation 

    If the right side has the form
    \(f(x)=e^{\alpha x}(P_1(x)cos \beta x+P_1(x)sin \beta x),\)
    then the particular solution \(y_p\)is given by 
    \(y_p=x^ke^{\alpha x}(R_1(x) cos \beta x +R_2(x)sin \beta x),\)
    Where the polynomials \(R_1(x)\) and \(R_2(x)\) have to be found by using the method of undetermined coefficients.
    *   If \(\alpha +\beta i\) is not a root of the characteristic equation, then the power k=0 ,
    *   If \(\alpha +\beta i\) is a simple root , then k=1 , 
    *   If \(\alpha +\beta i\) is a double root , then k=2.
  3. Differential Equations with Y Missing 
    \(y''= f(x,y').\)
    Set  u=y' . Then the new equation satisfied by v is 
    \(u'=f(x,u),\)
    Which is a first order differential equation. 
  4. Differential Equations with x Missing 
    \(y''=f(y,y').\)
    \(Set \;\;\;u=y'. Since\)
    \(y''={du \over dx }={du \over dy} {dy \over dx}=u{du \over dy },\)
    We have 
    \(u{du \over dy }= f(y,u),\)
    Which is a first order differential equation.
  5. Free Undamped Vibrations
    The motion of a Mass pn a Spring is described by the equation  \(m \ddot {y}+ky=0,\)
    where
    m is the mass of the object,
    k is the stiffness of the spring 
    y is displacement of the mass from equilibrium.

    The general solution is 
    \(y= A\, cos ( \omega_0t-\delta),\)
    Where
    A is the amplitude of the displacement ,
    \(\omega_0\) is the fundamental frequency, the peruod is \(T = {2 \pi \over \omega_0 },\)
    \(\delta\) is phase angle of the displacement ,
    This is and example of simple harmonic motion. 
  6. Free Damped Vibrations
    \(m \ddot{y}+\gamma \dot{y}+ky=0 ,\)Where
    \(\gamma\) is the damping coefficient. 
    There are 3 cases for the general solution :

    Case 1:   \(\gamma^2 > 4 km \) (overdamped )
    \(y(t)=Ae^{\gamma_1t}+Bc^{\gamma_2t},\)
    Where
    \(\lambda_1={- \gamma - {\sqrt {\gamma-4km}} \over 2 m},\)
    \(\lambda_2={- \gamma +{\sqrt {\gamma^2-4km}} \over 2 m},\)

    Case 2:  \(\gamma^2 > 4 km \) ( Crirically Damped )
    \(y(t)=(A+Bt)e^{\gamma t},\)
    Where
    \(\lambda =- {\gamma \over 2m}\).

    Case 3:  \(\gamma^2 > 4 km \) (underdamped )
    \(y(t)= e^{{\gamma \over2m}t} A\;cos(\omega t-\delta), \) Where 
    \(\omega=\sqrt {4km-\gamma^2} \; \; .\)
  7. Simple Pendulum 
    \({d^2 \theta \over dt^2}+{g \over L } \theta=0 ,\)
    where \(\theta\) is the angular displacement , L is the pendulum  length, g is the acceleration of gravity .
    \(\theta(t)=\theta_{max} sin \sqrt {{g \over L}t}, \) the period is \(T=2 \pi \sqrt {L \over g}\)   .
  8. RLC Circuit 
    \(L{d^2I \over dt^2}+R{dI \over dt}+{1 \over C}I=V'(t)=\omega E_0 \, cos (\omega t) ,\)
    Where I is the current in and RLC circuit with an ac voltage source \(V(t)= E_0 \; sin(\omega t).\)

    The general solution is 
    \(I(t)=C_1e^{r_1t}+C_2e^{r_2t}+A\, sin(\omega t-\phi),\)
    Where 
    \(r_{1,2}={-R \pm \sqrt {R^2-{4L \over C}} \over 2L} \, ,\)
    \(A= {\omega E_0 \over \sqrt {(L \omega^2-{1 \over C})}+R^2 \omega^2 } \; ,\)
    \(\phi= arctan ({L\omega \over R}-{1 \over RC\omega})\)
    \(C_1,C_2\) are constants depending on initial conditions.