Question

Integration factor

1. (x + 2y) dx + x dy = 0
2. ((x^2) + (y^2) + x) dx + y dy = 0
3. (5y - 6x) dx + x dy = 0
4. (3x - (y^2)) dx - 4xy dy = 0

Answer 1

Answer on Question #79308 – Math – Differential Equations Question

Integration factor

1. $(x + 2)dx + xdy = 0$

2. $(x^{2} + y^{2} + x)dx + ydy = 0$

3. $(5y - 6x)dx + xdy = 0$

4. $(3x - y^{2})dx - 4xydy = 0$

Solution

$$\begin{array}{l} M (x, y) d x + N (x, y) d y = 0 \\ \frac {\partial M}{\partial y} \neq \frac {\partial N}{\partial x} \\ \end{array}$$

Integration factor $u(x,y)$

$$\frac {\partial (u N (x , y))}{\partial x} = \frac {\partial (u M (x , y))}{\partial y}$$

1. $(x + 2)dx + xdy = 0$

$$M (x, y) = x + 2, \frac {\partial M}{\partial y} = 0$$ $$N (x, y) = x, \frac {\partial N}{\partial x} = 1$$ $$0 \neq 1 \Rightarrow \frac {\partial M}{\partial y} \neq \frac {\partial N}{\partial x}$$ $$\frac {\partial M}{\partial y} - \frac {\partial N}{\partial x} = \frac {0 - 1}{x} = - \frac {1}{x}$$

Then the integration factor $u(x)$ is function of $x$ only

$$u (x) = \exp \left(\int \frac {\frac {\partial M}{\partial y} - \frac {\partial N}{\partial x}}{N} d x\right) = \exp \left(\int \left(- \frac {1}{x}\right) d x\right) = \exp (- \ln | x |) = \frac {1}{x}$$

The integration factor

$$u = \frac {1}{x}$$

2. $(x^{2} + y^{2} + x)dx + ydy = 0$

$$M(x, y) = x^{2} + y^{2} + x, \quad \frac{\partial M}{\partial y} = 2y$$ $$N(x, y) = y, \quad \frac{\partial N}{\partial x} = 0$$ $$2y \neq 0 \Rightarrow \frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$$ $$\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x} = 2y - 0 = 2y$$ $$z = x^{2} + y^{2}, \quad \frac{\partial z}{\partial x} = 2x, \quad \frac{\partial z}{\partial y} = 2y$$ $$\frac{1}{u} \left(\frac{du}{dz}\right) = \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N \frac{\partial z}{\partial x} - M \frac{\partial z}{\partial y}} = \frac{2y}{y(2x) - (x^{2} + y^{2} + x)(2y)} = \frac{1}{x - x^{2} - y^{2} - x} = -\frac{1}{x^{2} + y^{2}} = -\frac{1}{z}$$ $$\frac{du}{u} = -\frac{dz}{z}$$ $$\int \frac{du}{u} = -\int \frac{dz}{z}$$ $$u = \frac{1}{z}$$

The integration factor

$$u = \frac{1}{x^{2} + y^{2}}$$

3. $(5y - 6x)dx + xdy = 0$

$$M(x, y) = 5y - 6x, \quad \frac{\partial M}{\partial y} = 5$$ $$N(x, y) = x, \quad \frac{\partial N}{\partial x} = 1$$ $$5 \neq 1 \Rightarrow \frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$$ $$\frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} = \frac{5 - 1}{x} = \frac{4}{x}$$

Then the integration factor $u(x)$ is function of $x$ only

$$u(x) = \exp \left(\int \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} dx\right) = \exp \left(\int \left(\frac{4}{x}\right) dx\right) = \exp (4 \ln |x|) = x^{4}$$ $$u(x) = x^4$$

4. $(3x - y^2)dx - 4xydy = 0$

$$M(x, y) = 3x - y^2, \quad \frac{\partial M}{\partial y} = -2y$$ $$N(x, y) = -4xy, \quad \frac{\partial N}{\partial x} = -4y$$ $$-2y \neq -4y \Rightarrow \frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$$ $$\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x} = -2y - (-4y) = 2y$$ \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}{N} = \frac{2y}{-4xy} = -\frac{1}{2x}

Then the integration factor $u(x)$ is function of $x$ only

$$u(x) = \exp \left( \int \frac{ \frac{\partial M}{\partial y} - \frac{\partial N}{\partial x} }{N} dx \right) = \exp \left( \int \left( -\frac{1}{2x} \right) dx \right) = \exp \left( -\frac{1}{2} \ln |x| \right) = \frac{1}{\sqrt{x}}$$

The integration factor

$$u = \frac{1}{\sqrt{x}}$$

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