Solving First Order Linear Differential Equations Using Variation of Parameters and Integrating Factors

When dealing with first order linear differential equations, two common methods are used: the method of variation of parameters (VOP) and the integrating factor method. This article will explore how to solve the differential equations using both methods and highlight key steps involved in each process.

Solving Using Variation of Parameters (VOP)

Consider the differential equation:

x_1y' - x_2y 5

First, we rearrange the equation in a standard form:

y' - frac{x_2}{x_1}y frac{5}{x_1}

To solve this using the method of variation of parameters (VOP), we set:

y u(x)v(x)

Then, we rewrite the differential equation using these substitutions:

u(x)v(x) u'(x)v(x) u(x)v'(x) - frac{x_2}{x_1}u(x)v(x) frac{5}{x_1}

Separating the homogeneous and inhomogeneous parts, we get:

u(x) left(v'(x) - frac{x_2}{x_1}v(x)right) frac{5}{x_1}

The homogeneous solution is found by solving:

v' - frac{x_2}{x_1}v 0

This is a separable differential equation:

int frac{1}{v} dv -int left(frac{x_2}{x_1}right) dx

ln|v| -ln|x_1| - x_2int frac{1}{x_1} dx

v frac{C}{|x_1|e^{x_2int frac{1}{x_1} dx|}}

For simplicity, let's denote:

v frac{1}{x_1 e^{x_2int frac{1}{x_1} dx}}

The particular solution is then:

u int frac{5}{x_1 v} dx 5x_1 e^{x_2int frac{1}{x_1} dx}

The general solution is given by:

y u(x)v(x) frac{5x_1 e^{x_2int frac{1}{x_1} dx}}{x_1 e^{x_2int frac{1}{x_1} dx}} frac{5}{x_1}

Solving Using Integrating Factor

Consider the differential equation:

x_1frac{dy}{dx} - x_2y 5x^{-1}

First, rewrite the equation in standard form:

frac{dy}{dx} - frac{x_2}{x_1}y 5x^{-1}

To solve this, we use the integrating factor method. The integrating factor is:

e^{int frac{x_2}{x_1} dx} e^{ln(x_1 x)} x_1 x

Multiplying the differential equation by the integrating factor:

x_1 x frac{dy}{dx} - x_1 x frac{x_2}{x_1}y 5x

x_1 x frac{dy}{dx} - x_1 x frac{x_2}{x_1}y 5x

frac{d}{dx}(x_1 x y) 5x

Integrating both sides:

x_1 x y int 5 x dx frac{5}{2}x^2 C

Therefore, the general solution is:

y frac{frac{5}{2}x^2 C}{x_1 x}

Conclusion

In conclusion, both the method of variation of parameters (VOP) and the integrating factor method provide powerful tools for solving first order linear differential equations. Understanding and applying these methods can greatly enhance one's problem-solving skills in mathematics and engineering.