Differential equations

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place in standard form.... See example.1(X)'' +/-#(X)'+/-#(X)=somethng Highest order X has a 1 in font of it.Also make sure that the variables are seperated you may need to use seperation of variable.

IVP/piecewise function/step function

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For these types of problems use Laplace.

First order

Can it be intergrsted directly.

Yes

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Intergrate the left side and the right side to get into the form Y= F(x)

No

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Use the intergrating factor technique. see hyperlink for tuturial.

Intergrating factor

Now that you have your intergrating factor, multiply both sides by it, integrate and solve for Y.

Second Order

Homoheneous

With constant coefficients

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Use real characteristic roots methos. found in section 4.2

With complex numbers

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Use complex charateristic roots method. found in section 4.3

Non-homogeneous

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For non-homogeneous problems us the follow methods.

Undermined coefficients

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This method is found in section 4.4

Variation of parameters

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found in section 4.5

Matrix

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For matrix and some systems use eigenvectors and eigenvalues to solve. If it can be put into a matrix format than use this.