# The method of undetermined coefficients can sometimes be used to solve first-order ordinary differential equations. Use the method to solve the following equations. y'-3y=5e^{3x}

Equations
The method of undetermined coefficients can sometimes be used to solve first-order ordinary differential equations. Use the method to solve the following equations.
$$y'-3y=5e^{3x}$$

2021-05-20

Step 1
The differential equation is $$y'-3y=5e^{3x}$$
Using the method of undetermined coefficient, the solution of the equation will be of the form $$y=y_{h}+y_{p}$$ where $$y_{h}$$ is the general solution to the corresponding homogeneous equation and $$y_{p}$$ is the particular solution.
Find the general solution to $$y'−3y=0$$.
The characteristic equation is $$r−3=0$$.
$$r-3=0$$
$$r=3$$
Thus, the general solution is, $$y_{h}=c_{1}e^{3x}$$.
Step 2
To find the particular solution, suppose $$y_{p}=Axe^{3x}$$.
$$y_{p}'=3Axe^{3x}-3Axe^{3x}=5e^{3x}$$
$$Ae^{3x}=5e^{3x}$$
$$A=5$$
Thus, the particular solution is $$y_{p}=5xe^{3x}$$.
Therefore, the solution of the equation is,
$$y=y_{h}+y_{p}$$
$$=c_{1}e^{3x}+5xe^{3x}$$
That is, $$y = c_{1}e^{3x}+5xe^{3x}$$.