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.
(a) [ \frac{dy}{dx}+xy=y^{2}e^{x^{2}/2}\sin x ] This is a Bernoulli equation. Divide by (y^{2}): [ y^{-2}\frac{dy}{dx}+x y^{-1}=e^{x^{2}/2}\sin x ] Let (v=y^{-1}). Then (\frac{dv}{dx}=-y^{-2}\frac{dy}{dx}). Hence [ -\frac{dv}{dx}+xv=e^{x^{2}/2}\sin x ] or [ \frac{dv}{dx}-xv=-e^{x^{2}/2}\sin x. _____ __________ ___ ____ _____ ____.
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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: .
b) Find the charge on the capacitor in an RLC circuit at sec. when
Henry,
ohms,
Farad.
.
c) Solve: .
Solve, using the method of variation of parameters
b) Solve the following equation by changing the independent variable
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) .
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.