Solve the following differential equations
(i) .
(ii) .
b) Find the equation of the integral surface of the differential equation
which passes through the line .
Let (p=\frac{\partial z}{\partial x},\ q=\frac{\partial z}{\partial y}). (i) Given (x^{2}p+y^{2}q=(x+y)z). This is a Lagrange type equation. The auxiliary system is [ \frac{dx}{x^{2}}=\frac{dy}{y^{2}}=\frac{dz}{(x+y)z}. ] From (\frac{dx}{x^{2}}=\frac{dy}{y^{2}}), [ \frac{dx}{x^{2}}-\frac{dy}{y^{2}}=0 \implies -\frac{1}{x}+\frac{1}{y}=C_{1}. ] Using (\frac{dx}{x^{2}}=\frac{dz}{(x+y)z}), [ \frac{dz}{z}=\frac{x+y}{x^{2}},dx=\left(\frac{1}{x}+\frac{y}{x^{2}}\right)dx. ] From the first integral, ( \frac{1}{y}=\frac{1}{x}+C_{1}), so (y=\frac{x}{1+C_{1}x}). Substituting, [ \frac{dz}{z}=\left(\frac{1}{x}+\frac{1}{x(1+C_{1}x)}\right)dx. _________ ____ ____ ______ ______ ____ _________.
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Solve: .
b) Find the charge on the capacitor in an RLC circuit at sec. when
Henry,
ohms,
Farad.
.
c) Solve: .
. 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 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.
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) .
b) Find the equation of the integral surface of the differential equation
which passes through the line .
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 differential equations
(i) .
(ii) .
(iii) .
b) Show that the wave equation can be reduced to the form
by the change of variable
.