MAT244--2019F > Quiz-2

tut 5103 quiz 2

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**Lan Cheng**:

show the given equation is not exact but becomes exact when multiplied by the given integrating factor, then solve the equation.

x^(2) * y^(3) + x * (1+y^2) * y' = 0, u(x, y) = 1/(xy^3).

First, let's show the given DE x^(2) * y^(3) + x * (1+y^2)y' = 0 is not exact.

Define M = x^(2) * y^(3), N = x * (1 + y^2).

M_y = d/(dy) [x^(2) * y^(3)] = 3x^(2) * y^(2)

N_x = d/(dx) [x * (1 + y^2)] = 1 + y^2

Since 3x^(2)y^(2) != 1 + y^2, the given DE is not exact.

multiply each side of given DE by integrating factor u, we get

1/(xy^3) * x^(2) * y^(3) + 1/(xy^3) * x * (1 + y^2)y' = x + (y^(-3) + y^(-1)) * y' = 0

Let the new M = 1/(xy^3) x^(2) * y^(3), new N = 1/(xy^3) * x * (1 + y^2)

M_y = 0, N_x = 0, M_y = N_x and the new DE is exact.

there exist \phy (x, y) such that

\phy x = M, \phy y = N.

\phy x = M = 1/(x * y^3) * x^(2) * y^(3)

Integrating both side by x, we have

\phy = 1/2 * x^2 + h(y)

\phy y = h'(y) = N = 1/(xy^3) * x * (1 + y^2)

h'(y) = 1/(xy^3) * x * (1 + y^2)

Integrating both side by y, we have

h(y) = -0.5 * y^(-2) + ln|y| + C

Altogether, we have

\phy (x,y) = 0.5 * x^2 - 0.5 * y^(-2) + ln|y| + C

So our general solution is

0.5 * x^2 - 0.5 * y^(-2) + ln|y| = C.

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