Solve the following differential equations
(i) .
(ii) .
(iii) .
b) Show that the wave equation can be reduced to the form
by the change of variable
.
(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 _____ ____ __________ _______ ____ _____ ________ ________.
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Solve the following differential equations
(i) .
(ii) .
b) Find the equation of the integral surface of the differential equation
which passes through the line .
Using the method of separation of variables, solve when
b) Find the temperature in a bar of length with both ends insulated and with initial temperature in the rod being
.
Solve the following differential equations
(i) .
(ii) .
(iii) .
b) Show that the wave equation can be reduced to the form
by the change of variable
.
. State whether the following statement are true or false. Justify your answer with the help of a short proof or a counter-example.
i) The initial value problem
has a unique solution in some interval of the form -h < x < h.
ii) The orthogonal trajectories of all the parabolas with vertices at the origin and foci on the x-axis is .
iii) The normal form of the differential equation
iv) The solution of the pde is
. (Note: The image appears to have a small typo in the solution provided, it should likely be to the power of -1 based on standard solutions, however, transcribing exactly as seen:
.)
v) The pde is hyperbolic in the entire xy-plane.
Solve, using the method of variation of parameters
b) Solve the following equation by changing the independent variable
Find the integrating factor of the differential equation
and hence solve it.
b) Solve the equation , for all positive integer values of m.
c) Solve the following IVP
Solve .
b) Write the ordinary differential equation
in the linear form, and hence find its solution.
c) Given that is one solution of the differential equation
find a second linearly independent solution of the equation.