Exercise 2 (4 marks) Consider the equation, (2y 6x)dr + (3r - 4xy 1)dy 0 1. Is it exact? 2. Use a special integrati...
1) Consider the following equation. 6xy dx (4y 9x2)dy-0 a) Show the equation is not exact. b) Find an integrating factor that will make it exact. c) Use the integrating factor to solve the resulting exact equation.
[8] 2. Consider the differential equation dx + (1 - sin(v)) dy = 0 Determine if the equation is exact. If so, solve. If not determine an approximation integrating acco the equation exact. Verify that the new equation is exact, and solve the differential equation using the integrating factor you have found. (Hint: the integrating factor should be a function of y only.)
a) Consider the first-order differential equation (y + cos.r) dx + dy = 0. By multiplying integrating factor y(x) = ei" to both sides, show that the differential equation is exact. Hence, solve the differential equation. (6 marks) b) Solve the differential equation (4.r + 5)2 + ytan z = dc COSC (7 marks)
Solve the equation (2x)dx + (2y - 4x2y-1)dy = 0 An implicit solution in the form F(x,y)=C is _______ =C, where is an arbitrary constant, and _______ by multiplying by the integrating factor.
number 5 please 1-14 ODES. INTEGRATING FACTORS Test for exactness. If exact, solve. If not, use an integrating factor as given or obtained by inspection or by the theorems in the text. Also, if an initial condition is given, find the corresponding particular solution. 1. 2xy dx + x2 dy = 0 2. xºdx + y°dy = 0 3. sin x cos y dx + cos x sin y dy = 0 4. €3°(dr + 3r de) = 0 5....
Find an integrating factor of the form X"y" and solve the equation. (2x-172-9y)dx + (3y-6x) dy=0, y(1) =1 OA 4x2y3 – 3x3y2 = 1 08.3x2y3 – x3y2=2 ocx?y* - 3x4y2 = -2 D.*?y3 - 3x3y2=-2 Ex?y2 – 3x3y2 = -2
4. Solve the exact differential equation. (1-2xy)dx + (4y3 - x2)dy 0 4. Solve the exact differential equation. (1-2xy)dx + (4y3 - x2)dy 0
Question 14 (12 marks) Consider the following separable differential equation. dy cos(z)(-1) dr (a) Find any constant solutions of this differential equation and hence write down the solution with initial value y=- when r=7 (b) Use partial fractions to evaluate 1 dy. 1 (c) Use the method for solving separable differential equations to solve this DE in the case where y 0 when r T. You may assume that the solution does not cross the constant solutions you found in...
Solve the equation. (2x)dx + (2y - 4x^y 'dy =0 by multiplying by the integrating factor. An implicit solution in the form F(x,y)=C is = C, where C is an arbitrary constant, and (Type an expression using x and y as the variables.) the solution y = 0 was lost the solution x = 0 was lost no solutions were lost
In this problem we consider an equation in differential form M dx + N dy = 0. The equation (2е' — (16х° уе* + 4e * sin(x))) dx + (2eY — 16х*y'е*)dy 3D 0 in differential form M dx + N dy = 0 is not exact. Indeed, we have For this exercise we can find an integrating factor which is a function of x alone since м.- N. N can be considered as a function of x alone. Namely...