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# Calculus and Analysis

### Differential Equation: Find General Solution

Please see the attached file for the fully formatted problem. Find a general solution on (-pi/2, pi/2) to y'' + y = tan x given that S secx dx = ln |sec x + tan x|.

### Undetermined coefficients

Use the method of undetermined coefficients to solve the attached differential equation (see attachment)

### Homogeneous Solutions and Differential Equations

Given that the differential equation y^n + p(x)y' + q(x)y = r(x) attached has three solutions of sin x, cos x and sin 2x. Find yh (yh is the corresponding homogeneous solution). See attachment for better formula representation. See the attached file.

### Wronskian Solution and Differential Equations

Given that the differential equation y^n + p(x)y' + q(x)y =0 has two solutions x^2 -x and x^3 - x. Use the Wronskian to find p(x). See attachment for better formula representation.

### Calculus Equation - Find a General Solution

Find a general solution to the equation in the attachment

### Finding and Solving the Initial Value Problem

Solve the initial value problem y" + 2y' + y = 0 y(0) = 1 y'(0) = -3. See the attached file.

### Calculus: Solving Using a General Solution

Find a general solution for the following equation. Make sure to show all of the steps: Equation: 3x^2y^"+ 11xy^' - 3y = 0, x > 0.

### Wronskians

Please see the attached file for the fully formatted problems. Determine whether the functions can be Wronskians on -1<x<1 for a pair of solutions to some equation y'' + py' +qy = 0 with p and q continuous. a) W(x) = 6e^4x b) W(x) = x^3 c) W(x) = 0 d) W(x) = (x -1/2)^1/2

### Important information about Linear or nonlinear

Determine whether the following questions are linear or nonlinear. (a) yy''-y'=sin(x) (b)x^2y''-y'+y=cos(x)

### Existence and uniqueness theorem

Determine the largest interval for which the existence and uniqueness theorem ensures a unique solution to See attached

### Vector Functions to Partial Derivative

Attached is more clear 1. Distance from a a point to a curve: Find the shortest distances between the point (1,2,1) and a point on the curve r(t)= (1/t*i)+(lnt(t)*j)+(sqrt(t)*k) 2. Distance from a point to a curve: Find the maxmium distances from the point (1,2,-1) to a point on the curve of intersection of the plane z=(

### Laplace transform and differential equations.

Find the inverse Laplace transform of (s^3+s^2+2/s) / [s^2(s^2+3s+2)] Using this (or otherwise), Find the solution of the equation y"+3y'+2y = 1-t Find the transform of the following functions: f(t) = (1+t^2)[u(t-1)-u(t-2)] where u(t) is the unit step function. f(t) = sin(t) for 0<t<Pi and f(t)=0 for Pi<t<2*Pi

### Non-Homogeneous Second Order Differential Equation

(d^2 *y)/(d*t^2) + 3*(dy/dt) + 2y = 24* exp(-4*t), y(0)=10, y'(0)=5.

See attachment.

### Diff. EQ

Solve for variable y in terms of t W/ given initial condition: dy/dt + 4y = 40sin3t y(0)=6

### Differential Equations

Solve y in terms of t with initial conditions given. a.) (d^2)y/dt^2+3dy/dt+2y=24e^-4t y(0)=10 y'(0)=5 b.) (d^2)y/dt^2+6dy/dt+9y=0 y(0)=10 y'(0)=0

### Convergent Series Natural Log Series

Use 1) sum of (x ^ (n +1))/ (n +1 ) converges uniformly on [-1, 0] 2) sum of x ^ n converges uniformly on (-1, 0] 3) sum of x ^ n = 1/(1-x) to show that ln 2 = 1 - 1/2 + 1/3 - 1/4 ...

View attachment

### Equation for Population P

View attachment for the problem.

### Use the Uniqueness Theorem for the Initial Value Problem

4. For the initial value problem dy/dx = 3y^(2/3), y(2) = 0, (a) does existence uniqueness Theorem 1 imply the existence of a unique solution? Explain. (b) Which of the following functions are solutions to the above differential equation? Explain. (b_1) y(x) = 0 (b_2) y(x) = (x - 2)^3 (b_3) y(x) = (x - alpha)^3, x <

### How do you find the inverse laplace transform

Find the inverse laplace transform: (L^-1) * [1 / (s^2 * (s^2 + 1))]

### How do you find the inverse laplace transform?

Find the inverse Laplace transform: (L-1) * [(e^-2s) / (s^2)]

### Inverse Laplace Transform

Please see the attached file for the fully formatted problems.

### Derivative Question - Algebraic Function

This is from a Trig/Calculus course...Explain FULLY: If F(x) = x^4 - 2x^3 + 4x^2 - 9 Note: ^ indicates exponant. Find F prime of x. It will be a derivative. I need every step explained clearly as I have a bet riding on this! I need to be able to show every step in order to win my bet.

### Differential Equation : General Solution

Please see the attached file for the fully formatted problems. Find a general solution on (-pi/2,pi/2) to y''+y=tan x given that S sec x dx = ln|secx + tanx|

### Differential Equations : Method of Undetermined Coefficients

Use the method of undetermined coefficients to solve the following differential equation. y'' + 2y' - 3y = 9x - 10 sin x y(0)=0 y'(0)=4

### Differential Equations : Method of Undetermined Coefficients

Decide whether the method of undertermined coefficients can be applied to find a particular solutions of the given equations. (Explain) a) y'' + 3y' - y = tan x b) y'' + xy' + y = sin x

### Find the solution (y sub p) to the following differential equation.

Y'' + 2y' - 3y = e^(-3x) + x^2 * e^x