2. Verify that the equation is not exact, compute the...

2. Verify that the equation is not exact, compute the integrating factor, verify that when the original equation is multiplied by the integrating factor that it is exact, and then derive the equation for g(x,y). a. (1 + y^2)dx + xydy = 0 b. (x - cos y)dx - sin y dy = 0 c. (xy^2 - 2y^3)dx + (3 - 2xy^2)dy = 0 d. (xy + 1)dx + x^2dy = 0 e. ydx + (x^2 + y^2 - x)dydy = 0 f. y' - 2y = t^2e^2t g. x dy/dx - 3y = x^4e^2x h. dy/dx + 2x/(1+x^2)y = 4/(1+x^2)

Transcript

00:01 For the differential equation 1 plus y square dx plus xy into dy equal to 0.

00:11 So comparing with comparing with mxy dx plus nx, y dy equal to 0 we have we have m of x, y equal to 1 plus y square and nx, y equal to xy.

00:37 Now find find del m divided by del y and del n divided by del x.

00:45 So del m divided by del y equal to del divided by del y of it's a derivative of m with respect to y that is 1 plus y square which is equal to 2y and del n divided by del x is the derivative of n with respect to x.

01:00 So del divided by del x into xy is equal to y.

01:04 Now since since del m divided by del y is not equal to del n divided by del x.

01:13 So the given differential equation is not exact given differential equation is not exact not exact.

01:23 So now now to make it now to make it exact to make it exact be multiplying be multiplying multiplying each part of the equation each part of the equation equation by say u equal to x to the power m and y to the power n.

01:50 So then we have so multiply by equation x to the power n to y to the power n implies x to the power n y to the power n plus x to the power m….y to the power n into y square into dx plus x to the power m y to the power n into xy into dy equal to 0.

02:11 Now after simplifying we have x to the power m y to the power n plus x to the power m into y to the power n plus 2 into dx plus x to the power m plus 1 into y to the power n plus 1 into dy equal to 0.

02:26 So now mxy equal to mx comma y equal to x to the power m y to the power n plus x to the power m into y to the power n plus 2 and nx comma y equal to x to the power m plus 1 into y to the power n plus 1...

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