# 4. Linear DEs of Order 1

If *P = P*(*x*) and *Q = Q*(*x*)
are functions of *x* only, then

`(dy)/(dx)+Py=Q`

is called a **linear differential equation order 1.**

We can solve these **linear** DEs using an **integrating
factor****.**

For linear DEs of order 1, the integrating factor is:

`e^(int P\ dx`

The **solution** for the DE is given by multiplying *y* by the integrating factor (on the left) and multiplying *Q *by the integrating factor (on the right) and integrating the right side with respect to *x*, as follows:

`ye^(intP\ dx)=int(Qe^(intP\ dx))dx+K`

### Example 1

Solve `(dy)/(dx)-3/xy=7`

Here is the solution graph of our answer for Example 1 (I've used *K* = 0.5).

It is a cubic polynomial curve:

Solution using `K=0.5`.

### Example 2

Solve `(dy)/(dx)+(cot\ x)y=cos\ x`

Here is the solution graph of our answer for Example 2 (I've used *K* = 0.1) .

It is a composite trigonometric curve, where the main shape is the cosecant curve, and the "wiggles" are due to the addition of the (sin *x*)/2 part:

Typical solution graph using `K=0.1`.

### Example 3

Solve `dy + 3y\ dx = e^(-3x)dx`

Here is the solution graph for Example 3 (I've used *K* = 5).

It was necessary to zoom out (a lot) to see what is going on in this graph.

Typical solution graph using `K=5`.

### Example 4

Solve `2(y - 4x^2)dx + x\ dy = 0`

Here is the solution graph for Example 4 (I've used *K* = 5).

There is a discontitnuity at *x* = 0.

Typical solution graph using `K=5`.

### Example 5

Solve `x(dy)/(dx)-4y=x^6e^x`

Here is the solution graph for Example 5 (I've used *K* = 0.005).

Typical solution graph using `K=0.005`.

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