Mathematics Solutions

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Homogeneous equations

Case 1 when roots are different and real:

*+

Case 2 when roots are equal and real:

*+

Case 3 when roots are complex:

* +

Case 4 when roots are equal and complex:

* + , -

Case 5 when roots are mixed:

*+

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Non Homogeneous equations

Case 6: on right side enx

occurs:

n is not root of auxiliary equation. Ycis found by aux equation. For yp

Put D = coefficient of power of x

*+

{ } ( )

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( )

*+

Case 7 on right side enx

occurs:

n is root of auxiliary equation Ycis found by aux equation. For yp

Polynomial is factorized.

*+

*+

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*+

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Case 8 on right side, occurs. is not root of differential equation. Ycis found by aux equation. For yp.

Trigonometric function is transformed into linear form. Put Substitution of is done until polynomial becomes linear. Rationalization is operated. Derivative is operated.

*+

Put

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Case 9 on right side, occurs. is not root of differential equation. Ycis found by aux equation. For yp.

Trigonometric function is transformed into linear form. is factorized. Use

o o

Then use previous cases Simplify Write down real/imaginary part of that statement.

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*+

Case 10 on right side, occurs. Ycis found by aux equation. For yp.

Put

Further steps according to last cases.

*+

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Case 11 At least one constant exists in differential equation.

Case 12 no constant exists.

On right side polynomial occurs i.e. Find by axillary equation. For .

Constant is taken as common. If constant doesntexist then least power derivative is taken

common.

Polynomial is transformed into . Binomial series is applied. Derivative is operated. Is operated.

( )

*+

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{ }

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Solve the differential equations for parallel lines:

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Solve the differential equations for non-parallel lines:

./ || ||

,- ,-

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,- ,-

|| ||

, - , -

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,- ,-

*+ *+

,-

, -

, -

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[ ] [ ]

BERNOLLIS EQUATION (Reduce able linear equation)

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,

-

,

-

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-

,

-

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