$$y\prime \left(x\right)+ay\left(x\right)=b$$

The general solution of the first order differential equation with constant coefficients is:

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

For b = 0 the homogeneous first-order linear differential equation with constant coefficients is available.

$$y\prime +ay=0$$

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 the constants:

The solution of the inhomogeneous differential equations can be obtained from the homogeneous one. Generally the solution of the inhomogeneous equation is given by the solution of the homogeneous equation plus a special solution of the inhomogeneous equation. The special solution can be obtained by the method of variation of constants. Here the constant C of the homogeneous solution is assumed as a function of x and the homogeneous solution is inserted into the inhomogeneous equation. C(x) is then determined so that the equation is fulfilled.

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

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 y_{s}

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

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

The calculator solves the initial value problem of y'+ay=b with the initial values x_{0}, y_{0}

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y'+ay=b y'+ay=ce^bx y'+2xy=xe^(-x^2) y'+xy=x y'+y=x y'=y^2 y'+y^2=1 y'=(Ay-a)(By-b)