22-06-2024 19:34

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Tags : Exact Linear Differential Equations Differential Equations

Equations Reducible to exact equations

Sometimes the differential equation which is not exact, can be made so on multiplication by a suitable factor is called integrating factor. The rules for finding integrating factors of the differential equation are as follows: $\mathrm{Solve}\left(x^2+y^2+2x\right)dx+2ydy=0.(\mathrm{ans};\cdot \left(\right))$

Methods

Here P is the function obtained ( of x or y)

• If $\frac{\frac{\partial M}{\partial y}-\frac{\partial N}{\partial x}}{N}=$ function of $x$ then $e^{\int Pdx}$ is the integrating factor
• If $\frac{\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}}{M}=$ function of $y$ then $e^{\int Pdy}$ is the integrating factor
• If the equation is in the form of $f_1\left(xy\right)ydx+f_2\left(xy\right)xdy=0$ Then integrating factor is $\frac{1}{Mx-Ny}$
• If the Differential Equation is Homogeneous Equation then its Integrating factor is $\frac{1}{Mx+Ny}$

References