57. Show that the substitution v = lny transforms the differential equation
dy/dx + P(x)y = Q(x)(ylny) into the linear equation dv/dx + P(x) = Q(x)v(x)

58. Use the idea in Problem 57 to solve the equation
x (dy/dx) - 4x2y + 2ylny = 0

59. Solve the differential equation dy/dx = (x-y-1)/(x+y+3) by finding h and k so that the substitutions x = u + h, y = v + k transform it into the homogeneous equation
dv/du = (u - v)/ (u + v)

60. Use the method in Problem 59 to solve the differential equation
dy/dx = (2y - x + 7) / (4x - 3y - 18)

keywords: ODE, ODEs

Solution Summary

Differential equations are solved. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

1. Complex Exponentials: Simply the following expression and give your answer both in polar and rectangular form.
o c=3ejπ/4+4e−jπ/2
2. Difference Equations: Solve the following difference equation using recursion by hand (for n=0 to n=4)
o y[n] + 0.5y[n-1] = 2x[n-1]; x[n] = δ[n], y[-1] = 0
3. DifferentialEquations

Differential Equation (IX): Formation of DifferentialEquations by Elimination
Eliminate the arbitrary constants from the equation: y = Ae^x + Be^2x + Ce^3x. Make sure to show all of the steps which are involved.

(1) Use Laplace Transforms to solve Differential Equation
y'' - 8y' + 20 y = t (e^t) , given that y(0) = 0 , y'(0) = 0
(2) Use Laplace Transforms to solve Differential Equation
y''' + 2y'' - y' - 2y = Sin 3t , given that y(0)=0 , y'(0)=0 ,y''(0)=0, y'''(0)=1
Note: To see the questions in their mathematic

Please see attached
Please show all work with explanations.
DifferentialEquations
5. Solve the differential equation with the given initial conditions:
6. Solve the differential equation with the given initial conditions:

(See attached file for full problem description)
1) The slope field for the system
dx/dt = 2x + 6y
dy/dt = 2x - 2y
is shown to the right
a) determine the type of the equilibrium point at the origin.
b) calculate all straight-line solution.
2) show that a matrix of the form A =(a b; -b a) with b!=0 must have complex eig

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.

Question 1.
1) Find a vector normal to the surface z + 2xy = x2 + y2 at the point (1,1,0).
2) Determine if there are separable differentialequations among the following ones and explain:
a) dy/dx=sin(xy),
b) dy/dx = (xy)/(X+y)
c) dr/d(theta) = (r^2+1)cos(theta)
3) Find the general solution of the differential