Question

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}

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asked 2021-05-19
The method of undetermined coefficients can sometimes be used to solve first-order ordinary differential equations. Use the method to solve the following equations.
\(y'-3y=5e^{3x}\)

Answers (1)

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, \(y = c_{1}e^{3x}+5xe^{3x}\).

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