1. Consider the following equations. (a) yi = B1 + B2x; + E; (b) y: = Bì + B2x; + E; (c) ŷ; = B1 + B2r; (d) yi = B1 + Bzx; + ê; (e) yi = ß1 + B2x; (f) yi = B1 + B2x; + ếi %3D ( s) Σει=0. (h) E; r¿ɛ; = 0. Which equations are correct or incorrect? Correct (g) and (h) if need be.

Convert the given equation into the form of a Sturm-Liouville equation. y" +7y' + ay = 0 Write the Sturm-Liouville form of the equation, using x as the independent variable.

Find a particular real solution for each of the following equations. The reasoning behindyour answers should be made clear. B.1. x'' + 4x' + 29x = 10e3t. B.2. u'' + 10u' + 169u = 50 cos(3t). B.3. v'''' + 13v'' + 36v = 10 cos(2t). These exercises are decoupled.

I already posted this problem but would like a more thorough explanation I am completely lost still.

In what way is the specific solution for the equation sought?

#1 Solve the following non-linear equations manually using: a) Bisection Method (5 iterations)

The following second-order equations explicitly contain y. Solve using the substitution v = y' (a) xy''+4y'=18x2

4. Verify whether the following equation is exact. If possible, solve the equation. a) (2xe*²+y² + 2xy)dx + (2ye"+y² + x²)dy = 0 b) (2xe+y² + 2xy)dx + (2ye"+y² +x)dy = 0 c) For both examples explain steps of your solution in your own words (line by line).

2. (Finding a second independent solution of an Equi-Dimensional (or Cauchy-Euler) equation with a double real root) In physical applications, we often encounter the Equi-Dimensional (or Cauchy-Euler) equation: ax'y"(x) + bxy'(x)+ cy(x) = 0 for constants a, b and c, which has solutions of the form y = x'. In the case x*y"(x) + (1– 2b)xy'(x)+b²y(x) = 0 for constant b, (2) we obtain only one value r = b by substitution, and the hunt is on for a second independent solution: a. Verify by substitution that y, = x° is a solution of the differential equation (2). Clearly state your conclusion. c. In order to obtain a second solution, assume that it is of the form y,«) = ux) y,) and determine the unknown function ux) by reduction of order. ulx) = so y,x) = d. Verify that y, and y, are linearly independent and state the interval of independence. Clearly state your conclusion.