Differential Equations Answers

Solve the following ODE using the power series method: (1 - X^2) y" - 2xy' + 2 y = 0

Solve the initial value problem: d^2x /dt^2 + 2 dy/dx - 3x = 0, X(2pie) = 1, x^.(2pie) = 13

Solve the following ordinary differential equations: I) ( 2y + X^2 + 1) dy /dx + 2 xy - 9x^2 = 0? 2) d^2y /dx^2 + 3*dy/dx + 2y = X^2

Obtain all the first and second order partial derivatives of the function: f ( x, y ) =ln ( 1 + xy ^ 2)

How to find maxima and minima. Process
1. to find x where dy/dx=o. let the x =a
2. then find d2y/dx2
3.if d2y/dx2 at x=a >0 then we will find minima for x=a
4.if d2y/dx2 at x=a <0 then we will find maxima for x=a
5.if d2y/dx2 at x=a =0 then we will have to proceed to the d3y/dx3
6.if d3y/dx3 at x=a in not eqal 0 , we will not find any maxima or minima.
7.if d3y/dx3 at x=a equals 0 then we will proceed to d4y/dx4. and if d4y/dx4 at x=a >0 then it's minima and if < 0 then maxima.

Please explain the rule 6 above. Why will we not find any maxima or minima in that case? Please explain with figure and example.

sin^-1 (-x)=??

sec^-1 (-x)=??

tan^-1(-x)=??

Cot^-1 (-x)=??

Cosec^-1 (-x)=??

explain in each case .

What is the differentiation of sec^-1 ((x^2+1)/(x^2-1))?

Why do we use x=0 to find f(0) in maclaurin series? why another numbers are not used?
and why f(0) is needed?

Find the solution of the Riccati equation:
dy/dx = 2cos^2 x - sin^2 x + y^2/ 2cosx ; y1(x) = sinx.

Find the value of b for which the equation
(y e^2xy + x) dx + bx e^2xy dy = 0
is exact, and hence solve it for that value of b .