In: Advanced Math

Diff. equations

Consider the equation (x^2 − 2)y''+ 3xy'+ y = 0.

a) Find the general solution as a power series centered at x = 0. Write the first six nonzero terms of the solution. And write the solution using sigma notation with a formula for the coefficients. Write the two linearly independent solutions that form the general solution.

b) Find a power series solution satisfying the initial conditions y(0) = 2 and y' (0) = 3. Write the first six nonzero terms of the solution. And write the solution using sigma notation with a formula for the coefficients.

Consider the following equation: (3 − x^2 )y'' − 3xy' − y = 0
Derive the general solution of the given differential equation
about x = 0. Your answer should include a general formula for the
coefficents.

Find two power series solutions of the given differential
equation about the point x=0
(x^2+2) y″+3xy'-y=0

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 of the following ODE: y''x^2 − 3xy' +
3y = 2/x , for x > 0.

y'(1+x^2)=(1/1-x)-3xy find the general solution.

Consider the equation y''− (sin x)y = 0.
Find the general solution as a power series centered at x = 0.
Write the first six nonzero terms of the solution. Write the two
linearly independent solutions that form the general solution.
Differential Equations

Find the general solution for the equations:
P(x) y"+ xy' - y = 0
a) P(x)= x
b) P(x)= x2
c) P(x) = 1

2. Find the general solution to the differential equation
x^2y''+ y'+y = 0 using the Method of Frobenius and power series
techniques.

Find
the general solution of the differential equations.
A:
cos(x)y′(x) +
y(x) sin(x) = 0
B:
2xy +
(x2 + y2)y′ = 0
C:
(4x3
+2xy2 +1)dx+(2x2y−1)dy=0.
D:
x(4x3
+2xy2 +1)dx+x(2x2y−1)dy=0
with
steps please.

solve using forbenious series
3xy''+(2-x)y'-y=0

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