Evaluate:
where Pn(x) is Legendre polynomial.
To evaluate the given integral numerically: \[ I_n(t) = \int_{-1}^{1} \frac{P_n(x)}{\sqrt{1 - 2xt + t^2}} \, dx \] where \( _____ ________ _________ ___ ___ ______ ________ ___ __________ _____ _________ ________.
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Show that the method.
is A-stable when applied to test equation y′ = λy, λ < 0 .
Solve the boundary value problem:
y"+y+f(x)=0
y'(0) = 0, y(1) = 0
by determining the appropriate Green’s function by using the method of variation of parameters and expressing the solution as a definite intergral.
Using method of Ferobenius, find the solution of the differential equation:
near x = 0.
Solve the differential equation:
x2y" +6xy'+(6+x2)y=0
in series about x = 0.
Solve the boundary value problem:
y′′ − 3y′ + 2y = 2
with
y(0) - y'(0) = -1
y(1) + y'(1) = 1
using the second order finite difference method with step length
Find the solution of = 0 in R subject to the boundary conditions:
where R is the square 0≤ x ≤1,0≤ y ≤1, using the five point formula. Use central difference approximation in the boundary condition. Assume uniform step length h = 1/2 along the axes.
Find the solution of the heat conduction equation subject to the given initial and boundary conditions:
Using Laasonen method with and .
Integrate for two levels.