1. Show that y1=cosx and y2=sinx are solution of the DE y''+y=0.verify that ⱷ(x)=c1cosx+c2sinx is also a solution of the DE.

2. Given that y1 and y2 are the solution of the DE y''+y^2=0. Is y3 =c1y1+c2y2 also a solution of the DE?

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- Apr 17th 2013, 10:16 PMcherrytreefriendSuperposition principle-homogeneous equation
1. Show that y1=cosx and y2=sinx are solution of the DE y''+y=0.verify that ⱷ(x)=c1cosx+c2sinx is also a solution of the DE.

2. Given that y1 and y2 are the solution of the DE y''+y^2=0. Is y3 =c1y1+c2y2 also a solution of the DE? - Apr 17th 2013, 10:19 PMMarkFLRe: Superposition principle-homogeneous equation
Just to let the helpers here know, this question has also been posted at MMF, and I have offered hints on how to proceed.

Quote:

For the first one, you will need to find $\displaystyle \varphi''(x)$ and substitute into the given ODE.

For the second one, you are told:

$\displaystyle y_1''+y_1^2=0$

$\displaystyle y_2''+y_2^2=0$

Multiply the first equation by $\displaystyle c_1$ and the second by $\displaystyle c_2$, then add the equations. What do you find?

- Apr 17th 2013, 10:21 PMcherrytreefriendRe: Superposition principle-homogeneous equation
wat is mmf?

- Apr 17th 2013, 10:24 PMMarkFLRe: Superposition principle-homogeneous equation
- Apr 17th 2013, 10:43 PMcherrytreefriendRe: Superposition principle-homogeneous equation
for the 2nd...i got until c1y1''+c1y1^2+c2y2"+c2y2^2 after adding two equation..

- Apr 17th 2013, 10:49 PMMarkFLRe: Superposition principle-homogeneous equation
What does this need to be equal to if $\displaystyle y_3(x)$ is a solution?

- Apr 17th 2013, 10:56 PMcherrytreefriendRe: Superposition principle-homogeneous equation
(c1+c2)y''+(c1+c2)y^2=0

- Apr 17th 2013, 11:02 PMMarkFLRe: Superposition principle-homogeneous equation
If $\displaystyle y_3\equiv c_1y_1+c_2y_2$ is a solution, then we must have:

$\displaystyle y_3''+y_3^2=0$

Substitute for $\displaystyle y_3$ into the ODE using the above definition, and see if this is equal to your earlier result... - Apr 17th 2013, 11:11 PMcherrytreefriendRe: Superposition principle-homogeneous equation
got it...thak you very much for guiding me...

- Apr 17th 2013, 11:16 PMMarkFLRe: Superposition principle-homogeneous equation
The purpose of these problems is to show you that the linear superposition of solutions only forms a solution for linear equations.