Solving First-Order Differential Equations

• A first-order Diff. Eq. In x and y is separable if it can be written so that all the y-terms are on one side and all the x-terms are on the other

First-Order Differential Equations• A differential equation has variables separable if it

is in one of the following forms:

• Integrating both sides, the general solution will be :

dy f(x)

dx g(y)= OR g(y)dy -

f(x)dx = 0

Cdxxfdyyg )()(

Separable Differential Equations

Another type separable differential equation can be expressed as the product of a function of x and a function of y.

dy g x h ydx

22dy xydx

Multiply both sides by dx and divide both sides by y2 to separate the variables. (Assume y2 is never zero.)

2 2 dy x dxy

2 2 y dy x dx

0h y

Example 1

Separable Differential Equations

Another type of separable differential equation can be expressed as the product of a function of x and a function of y.

dy g x h ydx

Example 1

22dy xydx

2 2 dy x dxy

2 2 y dy x dx

2 2 y dy x dx 1 2y x

21 xy

2

1 yx

2

1y Cx

0h y

Example 2

222 1 xdy x y edx

2

2

1 2 1

xdy x e dxy

Separable differential equation

2

2

1 2 1

xdy x e dxy

2u x2 du x dx

2

11

udy e duy

11 2tan uy C e C

211 2tan xy C e C

21tan xy e C Combined constants of integration

Example 2

222 1 xdy x y edx

21tan xy e C We now have y as an implicit function of x.

We can find y as an explicit function of x by taking the tangent of both sides.

21tan tan tan xy e C

2

tan xy e C

Notice that we can not factor out the constant C, because the distributive property does not work with tangent.

Example 3:Differential equation with initial condition – These are called Initial value problems

Solve the differential equation dy/dx = -x/y given the initial condition y(0) = 2.

• Rewrite the equation as ydy = -xdx• Integrate both sides & solve

• Since y(0) = 2, we get 4 + 0 = C, and therefore x2 + y2 = 4

xdxyyd

y2 + x2 = C where C = 2k

kxy 22

21

21

dxxfyg

dy

dxxfyg

dy

ygxfdxdy

)()(

)()(

)()(

Example 4

Solve: 2,1,12

yxwhenx

ydxdy

Solution to Example 4

cxyy

cxdyyy

xdx

ydy

dxxfyg

dy

ygxfx

ydxdy

ln11ln

21

ln1

11

121

1

)()(

)()(1

2

2

2

2

22

22

2

33

33

33

311

31ln

21

11ln

211ln2,1

xxy

xyxy

xyxy

xyy

c

yycyx