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
.
For the equation (u_{xt}=e^{-t}\cos x), integrate first with respect to (x): [ u_t=\int e^{-t}\cos x,dx=e^{-t}\sin x+f(t), ] where (f(t)) is an arbitrary function of (t). Integrating with respect to (t), [ u(x,t)=\int \big(e^{-t}\sin x+f(t)\big),dt=-e^{-t}\sin x+\int f(t),dt+g(x), ] where (g(x)) is an arbitrary function of (x). Using the initial condition _________ ________ __________ __________ ___ ________ ____ ____ _______.
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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 the following differential equations
(i) .
(ii) .
b) Find the equation of the integral surface of the differential equation
which passes through the line .
Solve, using the method of variation of parameters
b) Solve the following equation by changing the independent variable
Verify that the Pfaffian differential equation
is integrable and hence find its integral.
b) Solve the following equation by Jacobi's method
c) Show that , where a, b are arbitrary constants is a complete integral of
.
. 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 .
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.
Solve the following DEs
(i) .
(ii) .
b) The differential equation of a damped vibrating system under the action of an external periodic force is:
Show that, if n > m0 > 0 the complementary function of the differential equation represents vibrations which are soon damped out. Find the particular integral in terms of periodic functions.