Help with Particular integral

[Mechengstudent]

Hi, I am trying to figure out the particular integral of:

y''+4y'+3y = 8cosx-6sinx

The solution is: y(x)=Ae^-x+Be^-3x+2cosx+sinx

I have taken a look at the solution. However, I am unsure why it is 2cosx+sinx

I do not know why it is sinx instead of 2sinx, since the coefficient of the particular integral, a= 2e^x

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This is my current working out:

Complementary function u(x) = m²+4m+3 = 0

a = 1 b = 4 c = 3

-4+-sqrt(4²-4*1*3/2*1) = -1 and -3

So u(x) = Ae^-3x+Be^-x

For the particular integral u(x) = e^ax(Acos(x)+Bsin(x))

Substituting this expression:

m+4m+3m = e^ax(8cos(x)-6sin(x))

therefore a = 2e_x

Therefore, general solution is:

My answer ----> y=Ae^-3x+Be^-x+e^2x(8cos(x)-6sin(x))

Please let me know what I have done incorrectly.

 

[MarkFL]

Yes, your homogeneous solution is:

[MATH]y_h(x)=c_1e^{-x}+c_2e^{-3x}[/MATH]
Now, for the particular solution, (using the method of undetermined coefficients) you should assume it will take the form:

[MATH]y_p(x)=A\cos(x)+B\sin(x)[/MATH]
Hence:

[MATH]y_p'(x)=-A\sin(x)+B\cos(x)[/MATH]
[MATH]y_p''(x)=-A\cos(x)-B\sin(x)[/MATH]
Substituting into the ODE, we find:

[MATH](-A\cos(x)-B\sin(x))+4(-A\sin(x)+B\cos(x))+3(A\cos(x)+B\sin(x))=8\cos(x)-6\sin(x)[/MATH]
This simplifies to:

[MATH](A+2B)\cos(x)+(B-2A)\sin(x)=4\cos(x)-3\sin(x)[/MATH]
Equating like coefficients, we obtain the system:

[MATH]A+2B=4[/MATH]
[MATH]2A-B=3[/MATH]
Solving this system. we get:

[MATH](A,B)=(2,1)[/MATH]
Hence, our particular solution is:

[MATH]y_p(x)=2\cos(x)+\sin(x)[/MATH]
And so by superposition, the general solution to the ODE is:

[MATH]y(x)=y_h(x)+y_p(x)=c_1e^{-x}+c_2e^{-3x}+2\cos(x)+\sin(x)[/MATH]

 

[Mechengstudent]

Thanks MarkFL, great explanation.

 

[Steven G]

This is my current working out:

Complementary function u(x) = m²+4m+3 = 0

a = 1 b = 4 c = 3

-4+-sqrt(4²-4*1*3/2*1) = -1 and -3

You are in differential equations and can't (or don't want to?) think of two numbers that multiply out to 3 and add up to 4??

On top of that you wrote the quadratic formula wrong.