"File VPAR.MTH"

E_(x,i):=ELEMENT(x,i)

D_(x):=DIMENSION(x)

WRON(z):=VECTOR(DIF(z,x,k),k,0,D_(z)-1)

COL(g,n):=VECTOR(IF(k<n,0,g),k,n)

TACKON(x,v):=VECTOR(IF(k<D_(x)+1,E_(x`,k),v),k,D_(x)+1)`

MAT(z,g):=TACKON(WRON(z),COL(g,D_(z)))

VPAR(z,g):=z . INT(E_(ROW_REDUCE(MAT(z,g))`,D_(z)+1),x)

"####################"

"INSTRUCTIONS"

"If z is a vector of functions [y1, y2,...]"

"... then WRON(z) calculates the Wronskian."

"If z is a vector of solutions of a homogeneous linear equation"

"... and if g is the forcing function"

"... then MAT(z,g) gives the coefficient matrix"

"... of the system of equations that must be solved"

"... to execute variation of parameters."

"Under the conditions above, VPAR(z,g) executes"

"... variation of parameters and returns a particular solution."

"#######################"

"Example 1"

w:=[#e^x,x*#e^x,COS(x),SIN(x)]

"Simplify the following to see the Wronskian of w"

WRON(w)

"Example 2"

"Simplify the following to obtain a particular solution"

"... for y''''-2y'''+2y''-2y'+y="*SIN(x)^2

VPAR(w,SIN(x)^2)