What is the particular solution of the differntial equation {y}''+ 3{y}'+2y=4x+3?

Question:

What is the particular solution of the differntial equation {eq}{y}''+ 3{y}'+2y=4x+3 {/eq}?

Particular Solution:

The method of undetermined coefficients, the variation of parameters method are two types of methods for finding a particular solution of a non-homogeneous differential equation.

Consider a second order non-homogeneous differential eqaution {eq}{y}''+ a{y}'+by=g(x). {/eq}

In the method of undetermined coefficients, we choose a function that matches with the {eq}g(x). {/eq} if {eq}g(x) {/eq} is a polynomial function,then we choose a polynomial with unknown coefficients. If it is an exponential function we choose an exponential function.

In the method of variation of parameters, we use a direct formula for finding the particular solution.

We are asked to find the particular solution of {eq}{y}''+ 3{y}'+2y=4x+3. {/eq}

Particular solution depends on the function in the right hand side of the differential equation.

Here the function in the right-hand side is a polynomial of degree one. We are using the method of undetermined coefficients for finding the particular solution.

Hence choose {eq}y_p=ax+b. {/eq}

Then {eq}y_p'=a {/eq} and {eq}y_p''=0. {/eq}

Substituting {eq}y_p {/eq} and its derivatives in the given differential equation, we get:

{eq}{y_p}''+ 3{y_p}'+2y_p=4x+3 {/eq}

{eq}\Rightarrow 0+ 3a+2(ax+b)=4x+3 {/eq}

{eq}2ax+3a+2b=4x+3 {/eq}

Now comparing the coeffcients on both the sides, we get:

{eq}2a=4\Rightarrow a=2. {/eq}

{eq}3a+2b=3\\ \Rightarrow 3*2+2b=3\\ \Rightarrow 2b=-3 \Rightarrow \displaystyle b=\frac{-3}{2}. {/eq}

Hence the required particular solution is {eq}{\color{Blue} {\displaystyle y_p=2x-\frac{3}{2}}}. {/eq}