# Linear inhomogeneous differential equations of the 1st order Step-By-Step

$$7 y{\left(x \right)} + \frac{d}{d x} y{\left(x \right)} = \sin{\left(x \right)}$$

### Detail solution

Given the equation:$$7 y{\left(x \right)} + \frac{d}{d x} y{\left(x \right)} = \sin{\left(x \right)}$$

This differential equation has the form:

y' + P(x)y = Q(x)

where

$$P{\left(x \right)} = 7$$

and

$$Q{\left(x \right)} = \sin{\left(x \right)}$$

and it is called

**linear homogeneous**

**differential first-order equation:**

First of all, we should solve the correspondent linear homogeneous equation

y' + P(x)y = 0

with multiple variables

The equation is solved using following steps:

From y' + P(x)y = 0 you get

$$\frac{dy}{y} = - P{\left(x \right)} dx$$, if y is not equal to 0

$$\int \frac{1}{y}\, dy = - \int P{\left(x \right)}\, dx$$

$$\log{\left(\left|{y}\right| \right)} = - \int P{\left(x \right)}\, dx$$

Or,

$$\left|{y}\right| = e^{- \int P{\left(x \right)}\, dx}$$

Therefore,

$$y_{1} = e^{- \int P{\left(x \right)}\, dx}$$

$$y_{2} = - e^{- \int P{\left(x \right)}\, dx}$$

The expression indicates that it is necessary to find the integral:

$$\int P{\left(x \right)}\, dx$$

Because

$$P{\left(x \right)} = 7$$, then

$$\int P{\left(x \right)}\, dx$$ =

= $$\int 7\, dx = 7 x + Const$$

So, solution of the homogeneous linear equation:

$$y_{1} = e^{C_{1} - 7 x}$$

$$y_{2} = - e^{C_{2} - 7 x}$$

that leads to the correspondent solution

for any constant C, not equal to zero:

$$y = C e^{- 7 x}$$

We get a solution for the correspondent homogeneous equation

Now we should solve the inhomogeneous equation

y' + P(x)y = Q(x)

Use variation of parameters method

Now, consider C a function of x

$$y = C{\left(x \right)} e^{- 7 x}$$

And apply it in the original equation.

Using the rules

- for product differentiation;

- of composite functions derivative,

we find that

$$\frac{d}{d x} C{\left(x \right)} = Q{\left(x \right)} e^{\int P{\left(x \right)}\, dx}$$

Let use Q(x) and P(x) for this equation.

We get the first-order differential equation for C(x):

$$\frac{d}{d x} C{\left(x \right)} = e^{7 x} \sin{\left(x \right)}$$

So, C(x) =

$$\int e^{7 x} \sin{\left(x \right)}\, dx = \left(\frac{7 e^{7 x} \sin{\left(x \right)}}{50} - \frac{e^{7 x} \cos{\left(x \right)}}{50}\right) + Const$$

use C(x) at

$$y = C{\left(x \right)} e^{- 7 x}$$

and we get a definitive solution for y(x):

$$e^{- 7 x} \left(\frac{7 e^{7 x} \sin{\left(x \right)}}{50} - \frac{e^{7 x} \cos{\left(x \right)}}{50} + Const\right)$$