Question

Solve the following differential equations

 

(i) equation.

 

(ii) equation.

 

(iii) equation.
b) Show that the wave equation equation can be reduced to the form equation by the change of variable equation.

09 Jan 2026
Answer :
Word Count : 578
(i) Consider the differential equation: [ [D^3 - D D'^2 - D^2 + D D'] z = 0 ] Assume a solution of the form (z = e^{mx + ny}), so that (D \to m) and (D' \to n). Substituting gives the characteristic equation: [ m^3 - m n^2 - m^2 + m n = 0 ] Factor (m): [ m (m^2 - n^2 - m + n) = 0 ] So (m = 0) or (m^2 - m - n^2 + n = 0). Solving the quadratic in (m): [ m = \frac{1 \pm \sqrt{1 + 4n^2 - 4n}}{2} ] Hence the general solution is: [ z = f_1(y) + f_2(y) e^{\frac{1 + \sqrt{1 + 4n^2 - 4n}}{2} x} + f_3(y) e^{\frac{1 - \sqrt{1 + 4n^2 - 4n}}{2} x} ] where (f_1, f_2, f_3) are arbitrary functions of (y). --- (ii) Consider [ [D^4 - D'^4 - 2 D^2 D'^2] z = 0 ] Substitute
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