Differential Equations Reviewer PDF

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Differential Equations made simple. Summarized. Used for board exam review. ...

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DIFFERENTIAL EQUATIONS ELIMINATION OF ARBITRARY CONSTANTS Properties

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The order of differential equation is equal to the number of arbitrary constants in the given relation. The differential equation is consistent with the relation.

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The differential equation is free from arbitrary constants.

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Example Eliminate the arbitrary constants c1 and c2 from the relation → equation (1) → equation (2) → equation (3)

3 × equation (1) + equation (2) → equation (4)

3 × equation (2) + equation (3) → equation (5)

2 × equation (4) - equation (5)

answer

DIFFERENTIAL EQUATIONS OF ORDER ONE Separation of Variables Given the differential equation

Equation (1)

where and the form

may be functions of both and . If the above equation can be transformed into

Equation (2)

where is a function of alone and variables separable.

is a function of alone, equation (1) is called

To find the general solution of equation (1), simply equate the integral of equation (2) to a constant . Thus, the general solution is

Example 1.

, when

when

then,

,

,

answer 2. 3. 4.

, when , when

,

. ,

.

Equations with Homogeneous Coefficients If the function f(x, y) remains unchanged after replacing x by kx and y by ky, where k is a constant term, then f(x, y) is called a homogeneous function. A differential equation

Equation (1)

is homogeneous in x and y if M and N are homogeneous functions of the same degree in x and y.

To solve for Equation (1) let

or

Example 1.

Let

Substitute,

Divide by x2,

From

Thus,

answer

2. 3.

Exact Equations The differential equation

is an exact equation if

Steps in Solving an Exact Equation

1. Let

.

2. Write the equation in Step 1 into the form

and integrate it partially in terms of x holding y as constant. 3. Differentiate partially in terms of y the result in Step 2 holding x as constant. 4. Equate the result in Step 3 to N and collect similar terms. 5. Integrate the result in Step 4 with respect to y, holding x as constant. 6. Substitute the result in Step 5 to the result in Step 2 and equate the result to a constant. Example 1.

Test for exactness ; ; ; thus, exact! Step 1: Let

Step 2: Integrate partially with respect to x, holding y as constant

→ Equation (1) Step 3: Differentiate Equation (1) partially with respect to y, holding x as constant

Step 4: Equate the result of Step 3 to N and collect similar terms. Let...