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Mathematical problems for self assessment

Sample Problems
SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

 

1. Find the complementary function (and thus the general solution) of the following homogeneous second order linear differential equations:

(i) image placeholder

(ii) image placeholder

(iii) image placeholder

(iv) image placeholder

(v) image placeholder

[Ans. (i) image placeholder, (ii) image placeholder, (iii) image placeholder, (iv) image placeholder, (v) image placeholder].

 

2. Find the particular solution of the following homogeneous second order linear differential equations:

(i) image placeholder image placeholder image placeholder ,

(ii) image placeholder image placeholder .

[Ans. (i) image placeholder, (ii) image placeholder].

 

3. Find the general solution of the inhomogeneous second order linear differential equation,

image placeholder

[Ans. image placeholder].

 

4. Find the general solution of the inhomogeneous second order linear differential equation,

image placeholder

[Ans. image placeholder].

 

5. Find the particular solution of the inhomogeneous second order linear differential equation,

image placeholderimage placeholder.

[Ans. image placeholder].

 

6. Find the general solution of the inhomogeneous second order linear differential equation,

image placeholder

[Ans. image placeholder].

 

7. Find the particular solution of the inhomogeneous second order linear differential equation,

image placeholderimage placeholder

[Ans. image placeholder].

 

8. The charge, q, on a capacitor in an LCR circuit satisfies the second order differential equation

image placeholder,

where image placeholder and E are constants. Show that if image placeholderthen the general solution of the this equation is

If image placeholderand image placeholderwhen image placeholdershow that the instantaneous current in the circuit is given by

 

9. The displacement, x, of a periodically forced undamped harmonic oscillator satisfies the second order differential equation

image placeholder

where image placeholderis the undamped natural frequency of the oscillator and k and image placeholderare the amplitude and frequency respectively of the periodic force applied. Suppose image placeholder, then find the general solution to this equation in the cases:

(i) image placeholder,

(ii) image placeholder.

In case (ii) what happens to the oscillator as time t increases?

[Ans. (i) image placeholder(ii) image placeholder as t increases the displacement x evolves linearly with t and resonance occurs.].