Differential Equations Answers

The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem.
y = c1ex + c2e−x, (−∞, ∞); y'' − y = 0, y(0) = 0, y'(0) = 7

From the following data, find the number of students having weight between 60 and 70.
Weight (in kg) No.of students
0 - 40 250
40 - 60 120
60 - 80 100
80 - 100 70
100 - 120 50

Expand the function f(x)=P₃'(x) in a series of the form:
∞
∑ A_kP_k(x) ,where P₃(x)=1/2(5x³-3x)
k=0
The question is based on Legendre polynomials.

Show that: J_-1/2(x)=√(2/πx)cosx

Xp^2 - 2yp+ax=0

Under certain condition cane sugar in water is converted into dextrose at a rate which is proportional to the amount unconverted at any time. If of 75 gm, at time=0,89gm,are converted during the first 30 minutes,, the amount converted in one and half hours is

Using Rodrigue’s formula, obtain the expression for the Hermite polynomial H₄(x)

Show that:
i) J_1/2(x)=√(2/πx)cosx
ii) d/dx[xJ_1(x)]=xJ_0(x)

Using the method of Laplace transform, solve the following initial value problem:
y′′-3y′+2y=4e²ᵗ ;
y(0)=-3 ;
y′(0)=5 ;

Determine the Laplace transform of the function:
f(t)=
{cos(t-(2π/3)) t>2π/3
{ 0 t<2π/3