# Calculator first order differential equation

## Differential equation y'+ay=ce^bx

For a=b=c=1 follows y'+y=e^x

$y′(x)+ay(x)=cebx$

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

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

### Solution of the differential equation: y'+ay=ce^bx

In the first step the homogeneous equation has to be solved.

$y′+ay=0$

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

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

Integration

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

General solution of the homogeneous equation with undetermined constants k

Variation of the constants:

In a second step 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 k of the homogeneous solution is assumed as a function of x and the homogeneous solution is inserted into the inhomogeneous equation. k(x) is then determined so that the equation is fulfilled.

${y}_{h}^{\prime }=k\prime {e}^{-ax}-ak{e}^{-ax}$

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

$k\prime {e}^{-ax}-ak{e}^{-ax}$$+ak{e}^{-ax}$$=c{e}^{bx}$

Insertion into the inhomogeneous equation

$k\prime =c{e}^{\left(a+b\right)x}$

By rearranging we obtain an equation for the determination of k

$k=\frac{c}{a+b}{e}^{\left(a+b\right)x}$

Integration gives k(x)

${y}_{s}=\frac{c}{a+b}{e}^{bx}$

Insertion of k(x) in yh provides a special solution ys

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

This is the general solution of the inhomogeneous differential equation

### Calculator for the initial value problem of y'+ay=ce^bx

The calculator solves the initial value problem of y'+ay=ce^bx with the initial values x0, y0

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Initial values

x0=
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Parameter value

a=
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Axes ranges

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a-min=
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b-min=
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