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 ________ ____ _____ _______ __________ _____ ____.
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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 .
. 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 differential equations
(i) .
(ii) .
(iii) .
b) Show that the wave equation can be reduced to the form
by the change of variable
.
Solve, using the method of variation of parameters
b) Solve the following equation by changing the independent variable
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
.
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
.