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͖͔͕͙͖͔͗ —†ƒ‡–ƒŽ•‘ˆ

Definition

A differential equation is an equation that contains an unknown function and some of its derivatives.

The order of a differential equation is the order of the highest derivative that occurs in the equation.

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݀ݔ ൌ ʹݔ^ଷݕ ൅ ͳ

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݀ݔ^ଶ

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െ ͺݔ ݀ݕ

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ସ

൅ ͵ݔݕ ൌ ݔ െ ͳ

ן^ଶ ߲^ଶݑ

߲ݔ^ଶ ൌ ߲ݑ

߲ݐ

߲^ଷݑ

߲^ଶݔ߲ݐ ൌ ͳ ൅ ߲ݑ

߲ݐ

Ordinary Differential Equations (ODE)

Partial Differential Equations (PDE)

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െ ͺݔ ݀ݕ

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߲ݐ ߲^ଷݑ

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Solution of a differential equation

A function f is called a solution of a differential equation if the equation is satisfied when y = f(x) and its derivatives are

substituted into the equation.

If f is a solution of y' = xy, then ׊x א I, f'(x) = xf(x).

To solve a differential equation means finding all possible solutions of the equation.

The solution of y' = x is ^{ݕ ൌ ݔ}

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ʹ ൅ ܥǤ

Exercise

1. Show that every member of the family of functions

is a solution of the differential equation

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ͳ ൅ ܿ݁

ͳ െܿ݁

ݕԢ ൌ ݕ^ଶ െ ͳ ʹ Ǥ

Initial condition

Usually, we are not interested in a family of solutions (general solution). We want to find a particular solution that satisfies some additional requirement.

For example, it must satisfy a condition y(t₀) = y₀.

This is called an initial condition and the problem of finding a solution of this is called an initial-value problem.

Exercise

2. Find a solution of the differential equation that satisfies the initial condition y(0) = 2.

ݕԢ ൌ ݕ^ଶ െ ͳ ʹ

General and specific solution

Consider

then y = e

^2x

is a solution of this differential equation.

In addition, y = e

^x

+ e

^2x

is also one of the solution of this differential equation.

y

= Ce

^x

+ e

^2x

where C is a real number is a general solution of the differential equation.

y

= e

^x

+ e

^2x

is a specific solution for the differential equation.

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Exercise

3. Show that y = x – x^-1 is a solution of the differential equation xy' + y = 2x.

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Direction Fields

Sketch of solutions: Guides to sketch the graphs of solutions to the differential equation

y ′ = + x y

Long-term behavior: how the solutions behave as x increases.

Exercise

9. Sketch the direction field for ^݀ݕ_݀ݔ ^{ൌ › െ šǤ}

10. Sketch the direction field for ݕ^ᇱ ൌ ሺݕ^ଶ െ ݕ െ ʹሻሺͳ െ ݕሻ^ଶ.

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Separable equations

A separable equation is a first-order differential equation in which the expression y' can be factored as a function of x times the function of y. That means

We rewrite . It becomes

h(y) dy = g(x) dx

So that all y’s are on one side of the equation and all x’s are on the other side

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݄ ݕ ൌ ͳ

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Verification

We can verify that is the solution of

By using the Chain rule,

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݀ݔ ൌ ݃ ݔ

݄ ݕ ݀ݕ

݀ݔ ൌ ݃ ݔ

Exercise

4. Solve the differential equation and find the solution with the initial condition y(0) = 1.

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݀ݔ ൌ ݔ ݕ^ଶ

5. Solve the differential equation ^݀ݕ

݀ݔ ൌ ݁^ଶ௫ Ͷݕ^ଷǤ

Linear differential equation

A first order linear differential equation is one that can be put into the form

where P and Q are continuous functions on I.

Consider the first order linear differential equation xy' + y = 2x

LHS: xy' + y = (xy)' RHS:

Therefore, xy = x² + C.

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නʹݔ݀ݔ ൌ ʹݔ^ଶ ʹ ൅ ܥ

Exercise

6. Find the general solution of ݕԢ ൅ ͳ

ݔ ݕ ൌ ͵Ǥ

Integrating factor

To solve the first order linear differential equation, we multiply the suitable integrating factor, I(x).

to get

If we can find such a function I(x),

(I(x)y)' = I(x)Q(x).

Hence,

݀ݕ݀ݔ ൅ ܲ ݔ ݕ ൌ ܳ ݔ

ܫ ݔ ሺ݀ݕ

݀ݔ ൅ ܲ ݔ ݕሻ ൌ ܫ ݔ ܳ ݔ Ǥ

ܫ ݔ ݕ ൌ නܫ ݔ ܳ ݔ ݀ݔ ൅ ܥǤ

Integrating factor

The property of I(x) is

I(x)P(x) = I'(x).

This is a separable differential equation,

Hence,

න ͳ

ܫ ݔ ݀ܫ ݔ ൌ නܲ ݔ ݀ݔ

Ž פ ܫ ݔ פ ൅ܥ ൌ නܲ ݔ ݀ݔ ܫ ݔ ൌ ܣ݁^{׬ ௉ ௫ ௗ௫}Ǥ

Exercise

7. Solve the differential equation ^݀ݕ_݀ݔ ^{൅ ͵ݔଶݕ ൌ ͸ݔଶǤ}

8. Find the solution of the initial-value problem x²y' + xy = 1, x > 0 and y(1) = 2.

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—‘™•Šƒ•–Š‡‰‡‡”ƒŽˆ‘”ǣ

ƒέ  •›•–‡‘ˆŽ‹‡ƒ”‡“—ƒ–‹‘•

™Š‡”‡–Š‡…‘‡ˆˆ‹…‹‡–•ƒ_‹Œǯ•ǡƒ†‰_‹ ǯ•ƒ”‡ƒ”„‹–”ƒ”›ˆ—…–‹‘•‘ˆ–Ǥ

ˆ‡˜‡”›–‡”‰_‹ ‹•…‘•–ƒ–œ‡”‘ǡ–Š‡–Š‡•›•–‡‹••ƒ‹†–‘„‡Š‘‘‰‡‡‘—•Ǥ

–Š‡”™‹•‡ǡ‹–‹•ƒ‘Š‘‘‰‡‡‘—••›•–‡‹ˆ‡˜‡‘‡‘ˆ–Š‡‰ǯ•‹•‘œ‡”‘Ǥ

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‘”†‡”Ž‹‡ƒ”‡“—ƒ–‹‘•Ǥ

Ǧ–Š ‘”†‡”Ž‹‡ƒ”‡“—ƒ–‹‘ƒ†•›•–‡•‘ˆˆ‹”•–‘”†‡”Ž‹‡ƒ”‡“—ƒ–‹‘•

‘”‡šƒ’Ž‡ǡ

‹˜‡ƒǦ–Š ‘”†‡”Ž‹‡ƒ”‡“—ƒ–‹‘

ƒ‡–Š‡•—„•–‹–—–‹‘•ǣš͕ί›ǡš͖ί›ƍǡš͗ί›Ǝǡǥǡš ί›ȋΫ͕Ȍǡƒ†šƍ ί›ȋȌǤ

Š‡Ƥ”•–Ϋ͕‡“—ƒ–‹‘•ˆ‘ŽŽ‘™–Š—•Ž›Ǥƒ•–Ž›ǡ•—„•–‹–—–‡–Š‡šǯ•‹–‘–Š‡‘”‹‰‹ƒŽ

‡“—ƒ–‹‘–‘”‡™”‹–‡‹–‹–‘–Š‡Ǧ–Š ‡“—ƒ–‹‘ƒ†‘„–ƒ‹–Š‡•›•–‡‘ˆ–Š‡ˆ‘”ǣ

Exercise

Convert the following equation into a system of first order equations