For a=b=c=1 follows y'+y=e^x
The general solution of the first order differential equation with constant coefficients is:
In the first step the homogeneous equation has to be solved.
Solution of the homogeneous linear differential equation of first order with constant coefficients:
Transformation of equation
Division by y
Applying the chain rule
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.
Derivation of the homogeneous solution with k as a function of x
Insertion into the inhomogeneous equation
By rearranging we obtain an equation for the determination of k
Integration gives k(x)
Insertion of k(x) in yh provides a special solution ys
This is the general solution of the inhomogeneous differential equation
The calculator solves the initial value problem of y'+ay=ce^bx with the initial values x0, y0
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