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
.
Consider the Pfaffian form (M,dx+N,dy+P,dz=0) with (M=yz,; N=x^{2}y-zx,; P=x^{2}z-xy). The condition for integrability is (M\left(\frac{\partial N}{\partial z}-\frac{\partial P}{\partial y}\right)+N\left(\frac{\partial P}{\partial x}-\frac{\partial M}{\partial z}\right)+P\left(\frac{\partial M}{\partial y}-\frac{\partial N}{\partial x}\right)=0). Now (\frac{\partial N}{\partial z}=-x,; \frac{\partial P}{\partial y}=-x), hence the first bracket is zero. ________ __________ _______ _____ _______ _____ ______.
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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 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.
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
.
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
.
. 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.
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
.