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Linear Differential Equation

How to find the particular integral and complementary function of dydx−3y=2x+e4x\dfrac{dy}{dx} - 3y = 2x + e^{4x}?
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Avatar for mqb2766
mqb2766
21
Original post by esrever
How to find the particular integral and complementary function of dydx−3y=2x+e4x\dfrac{dy}{dx} - 3y = 2x + e^{4x}?


The complementary function is the homogeneous solution (i.e. the right hand side equals zero). It represents the response due to the initial condition, so
dydx−3y=0\dfrac{dy}{dx} - 3y = 0
You assume an exponential solution y=Aesty = Ae^{st} and solve for the auxiliary equation for s.

The particular intergral represents the "steady state" response due to the right hand side (the response when the initial conditions lie on the PI solution). There is a linear and an exponential term. The PI is of the form
B+Cx+De4xB + Cx + De^{4x}
You solve by finding the coeffs
http://epsassets.manchester.ac.uk/medialand/maths/helm/19_3.pdf

Note, you do not give an initial condition, so I'm presuming it is y(0)=0y(0) = 0.
Reply 2
Avatar for esrever
esrever
OP
20
Original post by mqb2766
The complementary function is the homogeneous solution (i.e. the right hand side equals zero). It represents the response due to the initial condition, so
dydx−3y=0\dfrac{dy}{dx} - 3y = 0
You assume an exponential solution y=Aesty = Ae^{st} and solve for the auxiliary equation for s.

The particular intergral represents the "steady state" response due to the right hand side (the response when the initial conditions lie on the PI solution). There is a linear and an exponential term. The PI is of the form
B+Cx+De4xB + Cx + De^{4x}
You solve by finding the coeffs
http://epsassets.manchester.ac.uk/medialand/maths/helm/19_3.pdf

Note, you do not give an initial condition, so I'm presuming it is y(0)=0y(0) = 0.


Thanks :biggrin: (PROSM)

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