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(x)=cebxa+b+ke-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=-ay

Transformation of equation

yy=-a

Division by y

(lny)=-a

Applying the chain rule

lny=-adx=-ax+k~

Integration

yh=ke-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.

yh=ke-ax-ake-ax

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

ke-ax-ake-ax+ake-ax=cebx

Insertion into the inhomogeneous equation

k=ce(a+b)x

By rearranging we obtain an equation for the determination of k

k=ca+be(a+b)x

Integration gives k(x)

ys=ca+bebx

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

y=ys+yh=ca+bebx+ke-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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Parameter value

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c=

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