In: Advanced Math

Consider the differential equation

(x

2 + 1)y

′′ − 4xy′ + 6y = 0.

(a) Determine all singular points and find a minimum value for the
radius of convergence of

a power series solution about x0 = 0.

(b) Use a power series expansion y(x) = ∑∞

n=0

anx

n

about the ordinary point x0 = 0, to find

a general solution to the above differential equation, showing all
necessary steps including the

following:

(i) recurrence relation;

(ii) determination of all coefficients in the power series;

(iii) final form of general solution as y(x) = c1y1 + c2y2.

Consider the differential equation:
y'(x)+3xy+y^2=0.
y(1)=0. h=0.1
Solve the differential equation to determine y(1.3)
using:
a. Euler Method
b. Second order Taylor series method
c. Second order Runge Kutta method
d. Fourth order Runge-Kutta method
e. Heun’s predictor corrector method
f. Midpoint method

Find the general solution for differential equation
x^3y'''-(3x^2)y''+6xy'-6y=0, y(1)=2, y'(1)=1,
y''(1)=-4

Consider the differential equation x′=[2 4
-2 −2],
with x(0)=[1 1]
Solve the differential equation where x=[x(t)y(t)].
x(t)=
y(t)=
please be as clear as possible especially when solving for c1
and c2 that's the part i need help the most

1 Let f(x, y) = 4xy, 0 < x < 1, 0 < y < 1, zero
elsewhere, be the joint probability density function(pdf) of X and
Y . Find P(0 < X < 1/2 , 1/4 < Y < 1) , P(X = Y ), and
P(X < Y ). Notice that P(X = Y ) would be the volume under the
surface f(x, y) = 4xy and above the line segment 0 < x = y <
1...

ﬁnd the general solution of the given differential equation.
1. y''+2y'−3y=0
2. 6y''−y'−y=0
3. y''+5y' =0
4. y''−9y'+9y=0

Given the differential equation y’’ +5y’+6y=te^t with start
value y(0) = 0 and y’(0). Let Y(s) be the Laplace transformed of
y(t).
a) Find an expression for Y(s)
b) Find the solution to the equation by using inverse Laplace
transform.

Solve the differential equation
y''+y'-2y=3, y(0)=2, y'(0) = -1

(Differential Equations) Consider the differential equation
xy’-x4y3+y=0
Verify that the function y =
(Cx2-x4)-1/2 is a solution of the
differential equation where C is an arbitrary constant.
Find the value of C such that y(-1) = 1. State the solution of
the initial value problem.
State the interval of existence.

For
the differential equation (2 -x^4)y" + (2*x -4)y' + (2*x^2)y=0.
Compute the recursion formula for the coefficients of the power
series solution centered at x(0)=0 and use it to compute the first
three nonzero terms of the solution with y(0)= 12 , y'(0) =0

Find the infinite series solution about x = 0 for the following
differential equation x2y"+ 4xy' + (2+x)y = 0,without using k
substitution and using Bessel's, Legrende's, or frobenius
equations.

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