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September 19th, 2019, 11:44 AM  #1 
Senior Member Joined: Dec 2015 From: Earth Posts: 826 Thanks: 113 Math Focus: Elementary Math  Nonorder DE
Solve equation : $\displaystyle y''=y+e^{2x}$ .

September 19th, 2019, 01:14 PM  #2 
Senior Member Joined: Jun 2019 From: USA Posts: 380 Thanks: 205 
$y = \Sigma c_n e^{\lambda_n x}$ $\lambda = 2$ or $\lambda^2  1 = 0$ $\rightarrow y=\frac{1}{3} e^{2x} + c_1 e^x + c_2 e^{x}$ $c_1, ~c_2$ from initial/boundary conditions 
September 19th, 2019, 08:28 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 21,110 Thanks: 2326 
$e^{x}y''  e^{x}y = e^x \\ e^{x}y' + e^{x}y = e^x + 2\text{A} \\ e^xy' + e^xy = e^{3x} + 2\text{A}e^{2x} \\ e^xy = \frac13e^{3x} + \text{A}e^{2x} + \text{B} \\ y = \frac13e^{2x} + \text{A}e^x + \text{B}e^{x}$ 
September 21st, 2019, 05:48 AM  #4 
Senior Member Joined: Dec 2015 From: Earth Posts: 826 Thanks: 113 Math Focus: Elementary Math 
Looks similiar to Wronskian . $\displaystyle dW(y,e^{x} )=e^{x}dx$.

September 21st, 2019, 05:28 PM  #5  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,697 Thanks: 2681 Math Focus: Mainly analysis and algebra  Quote:
The more standard approach would be to solve the characteristic polynomial for $y''y=0$ ($r=\pm1$) and them find a particular solution for the original equation, probably using the method of undetermined coefficients with $y_p=Ae^{2x}$.  

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