# Linear first order differential equation

## Differential equations

Differential equations are equations between unknown functions and their variables, and the derivatives of the unknown functions.

## Ordinary differential equations (ODE)

Ordinary differential equations are equations between unknown functions of one variable and the derivatives of the unknown functions.

## Partial differential equations

A partial differential equations is when the unknown function of more than one variable depends.

## Differential equations

Differential equations are equations which contains a function and derivatives of this function.

## Order

The order of the differential equations is the highest derivative of the function appearing in the equation. A first order differential equation thus includes the function and a maximum of the first derivative of the function.

## Notations for derivatives

$d y d x = d d x yx = y′x = y′$

## Direction field of the differential equation

Explicit ODE defined for each point of the XY plane, the gradient of the solution of the differential equation which passes through this point. Are put on a grid of the x / y plane tangents for the slope in the lattice point are obtained on the field direction. From the direction field one can estimate the function curve for different initial values ​​of the solution. The equation for the field direction is obtained by transforming the differential equation.

$y′=fxy$

## Initial values

The general solution of the differential equation is a family of functions parameterized by the constant C. The determination on a specific solution by specifying initial values​​. In order to specify a function value y 0 meant at x 0 . This can be determined from the general solution, the constant C.

${y}_{0}=\frac{b}{a}+C{e}^{-a{x}_{0}}$

The constant C is determined by:

$C=\left({y}_{0}-\frac{b}{a}\right){e}^{a{x}_{0}}$

## Linear differential equation first order with constant coefficients

Eine der einfachsten Differentialgleichungen ist die lineare Differentialgleichung erster Ordnung mit konstanten Koeffizienten.

$y′+ay=b$

$\text{with}\phantom{\rule{1em}{0ex}}a,b\phantom{\rule{0.5em}{0ex}}\in \phantom{\rule{0.5em}{0ex}}\mathbb{R}$

For b = 0 the homogeneous linear differential equation of first order with constant coefficients exists.

$y′+ay=0$

## Solution of the homogeneous linear differential equation of first order with constant coefficients

$y\prime =-ay$

Transformation of equation

$\frac{y\prime }{y}=-a$

Division by y

$\left(\mathrm{ln}y\right)\prime =-a$

Applying the chain rule

$\mathrm{ln}y=-a\int \mathrm{dx}=-ax+\stackrel{~}{C}$

Integration

${y}_{h}=C{e}^{-ax}$

General solution of the homogeneous equation with undetermined constants C

## Variation of constants

The solution of the inhomogeneous differential equations can be obtained from the homogeneous. In general, the solution to the inhomogeneous equation is given by the solution of the homogeneous equation plus a special solution to the inhomogeneous equation. The specific solution can be determined by the method of the variation of the constants. The constant C, the homogeneous solution is believed used as a function of x and the homogeneous solution to the inhomogeneous equation. C (x) is then determined so that the equation is satisfied.

${y}_{h}^{\prime }=C\prime {e}^{-ax}-aC{e}^{-ax}$

Derivation of the homogeneous solution with C as a function of x

$C\prime {e}^{-ax}-aC{e}^{-ax}$$-aC{e}^{+ax}$$=b$

Insertion into the inhomogeneous equation

$C\prime =b{e}^{ax}$

By rearranging we obtain an equation for the determination of C

$C=\frac{b}{a}{e}^{ax}$

Integration gives C(x)

${y}_{s}=\frac{b}{a}$

Insertion of C (x) in y h provides a special solution ys

$y={y}_{s}+{y}_{h}=\frac{b}{a}+C{e}^{-ax}$

This is the general solution of the inhomogeneous differential equation with constant coefficients

## Calculator for the linear first order differential equation with constant coefficients

$y=\frac{b}{a}+C{e}^{-ax}$

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a-min= a-max=
b-min= b-max=
a= b=
x0= y0=

## Linear first order differential equation

The general linear first order differential equations is given as follows.

$y′+fxy=gx$

The general solution is obtained by the following formula.

$yx=1e∫fxdx∫gxe∫fxdxdx+C$

## Calculator for the linear differential equation of first order

$y′+fxy=gx$

y' + * y =
step=
x-min= x-max=
y-min= y-max=
x0= y0=