First Order Differential Equations - Lecture Slides | MATH 114, Study notes of Mathematics

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Math 114-004, Fall 2009

Tong Zhu

Department of Mathematics University of Pennsylvania

September 15, 2009

First Order Differential Equations

First Order Differential Equations

I (^) Two approximation methods:

1. Direction fields
2. Euler’s method

First Order Differential Equations

I (^) Two approximation methods:

1. Direction fields
2. Euler’s method

I (^) Two simple explicitly-solvable equations:

1. Separable differential equations
2. Linear equations

First-Order Linear Differential Equations

First-Order Linear Differential Equations

a(x) dy dx

• b(x)y = c(x)

First-Order Linear Differential Equations

a(x) dy dx

• b(x)y = c(x)

Remark: it is linear in the dependent variable y (y and its derivative only occur to the first power, whose coefficients are functions of x alone).

First-Order Linear Differential Equations

a(x) dy dx

• b(x)y = c(x)

Remark: it is linear in the dependent variable y (y and its derivative only occur to the first power, whose coefficients are functions of x alone).

The standard form:

dy dx

• P(x)y = Q(x),

where P(x) and Q(x) are continuous on a given interval.

First-Order Linear Differential Equations

a(x) dy dx

• b(x)y = c(x)

Remark: it is linear in the dependent variable y (y and its derivative only occur to the first power, whose coefficients are functions of x alone).

The standard form:

dy dx

• P(x)y = Q(x),

where P(x) and Q(x) are continuous on a given interval.

Important: you should write first-order differential equations in the standard form before solving it with the following method.

How to solve this equation?

How to solve this equation?

I (^) It is not separable. But we need to get an equation whose one side is the derivative of something, and the other side is a function of one variable. So that we can integrate both sides to cancel out the derivative.

How to solve this equation?

I (^) It is not separable. But we need to get an equation whose one side is the derivative of something, and the other side is a function of one variable. So that we can integrate both sides to cancel out the derivative. I (^) Recall: the “product rule of derivatives”

(f (x) · g (x))′^ = f ′(x) · g (x) + f (x) · g ′(x)

I (^) Idea: multiply a factor, say I (x), on both sides of the linear equation

I (x)(y ′^ + P(x)y ) = I (x)Q(x) I (x)y ′^ + (I (x)P(x))y = I (x)Q(x)

I (^) Idea: multiply a factor, say I (x), on both sides of the linear equation

I (x)(y ′^ + P(x)y ) = I (x)Q(x) I (x)y ′^ + (I (x)P(x))y = I (x)Q(x)

Observe the two sides. If I (x)P(x) = I ′(x), then the above equation becomes

(I (x)y )′^ = I (x)Q(x)

Solvable!

I (^) Idea: multiply a factor, say I (x), on both sides of the linear equation

I (x)(y ′^ + P(x)y ) = I (x)Q(x) I (x)y ′^ + (I (x)P(x))y = I (x)Q(x)

Observe the two sides. If I (x)P(x) = I ′(x), then the above equation becomes

(I (x)y )′^ = I (x)Q(x)

Solvable!

I (x)y =

∫

I (x)Q(x)dx ⇒ y (x) =

I (x)

∫

I (x)Q(x)dx + C

I (^) Idea: multiply a factor, say I (x), on both sides of the linear equation

I (x)(y ′^ + P(x)y ) = I (x)Q(x) I (x)y ′^ + (I (x)P(x))y = I (x)Q(x)

Observe the two sides. If I (x)P(x) = I ′(x), then the above equation becomes

(I (x)y )′^ = I (x)Q(x)

Solvable!

I (x)y =

∫

I (x)Q(x)dx ⇒ y (x) =

I (x)

∫

I (x)Q(x)dx + C

Such an I (x) is called an integrating factor.

How to find such an I (x)?

How to find such an I (x)?

From above, we need

I (x)P(x) = I ′(x) = dI dx P(x)dx =

I

dI ∫ 1 I

dI =

∫

P(x)dx

ln |I (x)| =

∫

P(x)dx ⇒

I (x) = Ae

R (^) P(x)dx ,

where A = ±eC^.

How to find such an I (x)?

From above, we need

I (x)P(x) = I ′(x) = dI dx P(x)dx =

I

dI ∫ 1 I

dI =

∫

P(x)dx

ln |I (x)| =

∫

P(x)dx ⇒

I (x) = Ae

R (^) P(x)dx ,

where A = ±eC^. Here we only need one integrating factor. So we may take A = 1 and I (x) = e

R (^) P(x)dx

Solution Process

Solution Process

1. Write the linear equation in the standard form y ′^ + P(x)y = Q(x).

Solution Process

1. Write the linear equation in the standard form y ′^ + P(x)y = Q(x).
2. Find the integrating factor I (x) = e

R (^) P(x)dx .

Solution Process

1. Write the linear equation in the standard form y ′^ + P(x)y = Q(x).
2. Find the integrating factor I (x) = e

R (^) P(x)dx .

3. Multiply I (x) on both sides of the linear equation. And verify that the left hand side is just (I (x)y )′.

Solution Process

1. Write the linear equation in the standard form y ′^ + P(x)y = Q(x).
2. Find the integrating factor I (x) = e

R (^) P(x)dx .

3. Multiply I (x) on both sides of the linear equation. And verify that the left hand side is just (I (x)y )′.
4. Integrate both sides. DON’T FORGET A CONSTANT TERM.

Solution Process

1. Write the linear equation in the standard form y ′^ + P(x)y = Q(x).
2. Find the integrating factor I (x) = e

R (^) P(x)dx .

3. Multiply I (x) on both sides of the linear equation. And verify that the left hand side is just (I (x)y )′.
4. Integrate both sides. DON’T FORGET A CONSTANT TERM.
5. Solve for y (x).