Question

Using Taylor series expansion method; find a series solution of the initial value problem

(x^{2}+1)d^{2}y/dx^{2}+xdy/dx+2xy=0
y(0)=2 y'(0)=1

Answer #1

1) Consider the following initial-value problem.
(x + y)2 dx + (2xy + x2 − 2) dy =
0, y(1) = 1
Let af/ax = (x + y)2 = x2 + 2xy +
y2.
Integrate each term of this partial derivative with respect to
x, letting h(y)
be an unknown function in y.
f(x, y) = + h(y)
Find the derivative of h(y).
h′(y) =
Solve the given initial-value problem.
2) Solve the given initial-value problem.
(6y + 2t − 3)
dt...

method of separation of variables.(x2y+ 2xy+ 5y)dy=
(y2+ 7)dx2
integration factor. xdy/dx−2y=x4sinx3.
variation of parameters: y′−y=xex/x+1

1)Consider the following initial-value problem.
(x + y)2 dx + (2xy + x2 − 2) dy =
0, y(1) = 1. Let af/ax = (x + y)2 =
x2 + 2xy + y2. Integrate each term of this
partial derivative with respect to x, letting
h(y) be an unknown function in y.
f(x, y) = + h(y)
Solve the given initial-value problem.
2) Solve the given initial-value problem.
(6y + 2t − 3)
dt + (8y + 6t
− 1) dy...

Find the first four terms in the Taylor series expansion of the
solution to
y′(x) = 2xy(x)−x3, y(0) = 1.

Solve the given initial-value problem. The DE is a Bernoulli
equation.
x2 dy/dx-2xy=5y^4, y(1)=1/2

Transform the differential equation x2d2y/
dx2 − xdy/dx − 3y = x 1−n ln(x), x > 0 to
a linear differential equation with constant coefficients. Hence,
find its complete solution using the D-operator method.

Solve the initial-value problem.
(x2 + 1)
dy
dx
+ 3x(y − 1) = 0,
y(0) = 4

Find the
i)particular integral of the following differential equation
d2y/dx2+y=(x+1)sinx
ii)the complete solution of d3y /dx3-
6d2y/dx2 +12 dy/dx-8 y=e2x
(x+1)

Use the power series method to find the solution of the initial
value problem. Write the first eight nonzero terms of the power
series centered at x = 0.
y′′= e^y, y(0) = 0, y′(0) = −1

7) (1 − x2)y'' − 2xy' + 2y = 0; y(0) = 0, y'(0) =
3
Find the power series solution about x=0.

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