a) Solve the following ordinary differential equations:
i)
ii)
b) Solve the initial value problem:
I will solve these ordinary differential equations (ODEs) numerically using Python. The numerical approach is useful when analytical solutions are difficult to obtain. I'll use numerical integration techniques such as Euler's method or `scipy.integrate.solve_ivp` for solving these equations. __________ __________ _______ ___ _______ ________ ___.
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a) Solve the following ordinary differential equations:
i)
ii)
b) Solve the initial value problem:
2. An emf of 100 V is applied to a series RC circuit in which the resistance is 200 ohms
and the capacitance is farads. Determine the charge q(t) on the capacitor if q(0) = 0. Also determine the current i(t).
Reduce the following PDE to a set of three ODEs by the method of separation of variables.
Obtain the Fourier series for the following periodic function which has a period of 2
for
b) Obtain all the first and second order partial derivatives of the function:
The electrostatic potential in a rectangular region is governed by the two
dimensional Laplace equation:
and
Use the Frobenius method to obtain one solution of the following ODE:
a) Show that the function is a solution of the one-dimensional wave equation.
ii) sin
is a solution of the one-dimensional heat equation.