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REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... % Tests and demonstrations for the odesolve package % First some tests of the testdf module algebraic procedure showode(); <<write "order is ", odeorder, " and degree is ", odedegree; write "linearity is ", odelinearity," and highestderiv is ", highestderiv>>; showode depend y,x$ ode1 := df(y,x); ode1 := df(y,x) sortoutode(ode1, y, x)$ showode()$ order is 1 and degree is 1 linearity is 1 and highestderiv is df(y,x) sortoutode(ode1**2,y,x)$ showode() $ order is 1 and degree is 2 linearity is 2 and highestderiv is df(y,x) sortoutode(e**ode1,y,x) $ showode() $ order is 1 and degree is 0 df(y,x) linearity is e and highestderiv is df(y,x) sortoutode(df(y,x)*df(y,x,2),y,x) $ showode() $ order is 2 and degree is 1 linearity is 2 and highestderiv is df(y,x,2) nodepend y,x $ depend z,w $ sortoutode(df(z,w,2)+3*z*df(z,w)+e**z,z,w) $ showode() $ order is 2 and degree is 1 z linearity is e and highestderiv is df(z,w,2) nodepend z,w $ % ****************************************** % Next some tests for first-order differential equations depend y,x $ % Just to test tracing on trode $ % First-order quadrature case ode := df(y,x) - x**2 - e**x; x 2 ode := df(y,x) - e - x odesolve(ode, y, x); This first-order ODE can be solved by quadrature x 3 3*arbconst(1) + 3*e + x {y=---------------------------} 3 % A first-order linear equation, with an initial condition ode:=df(y,x) + y * sin x/cos x - 1/cos x; cos(x)*df(y,x) + sin(x)*y - 1 ode := ------------------------------- cos(x) ans:=odesolve(ode,y,x); This is a first-order linear ODE solved by the integrating factor method ans := {y=arbconst(2)*cos(x) + sin(x)} % Note that arbconst is declared as an operator % The initial condition is y = 1 at x = 0 so we do... arbconst(!!arbconst) := sub(y=1,x=0,rhs first solve(ans,arbconst(!!arbconst))); arbconst(2) := 1 ans; {y=cos(x) + sin(x)} clear arbconst(!!arbconst) $ % A simple separable case ans := odesolve(df(y,x) - y**2,y,x); This is a first-order separable ODE arbconst(3)*y - x*y - 1 ans := {-------------------------=0} y % We can improve this by solve(ans,y); 1 {y=-----------------} arbconst(3) - x nodepend y,x $ % A separable case, in different variables, with an initial condition depend z,w $ ode:= (1-z**2)*w*df(z,w)+(1+w**2)*z; 2 2 ode := - df(z,w)*w*z + df(z,w)*w + w *z + z % Assign the answer so we can input the condition (z = 2 at w = 1/2) ans:=odesolve(ode,z,w); This is a first-order separable ODE 2 2 2*arbconst(4) - 2*log(w) - 2*log(z) - w + z ans := {-----------------------------------------------=0} 2 % To tidy up the answer we will get for the constant we use for all x let log(x)+log(1/x)=0 $ arbconst(!!arbconst) := sub(z=2,w=1/2, rhs first solve(ans,arbconst(!!arbconst))); - 15 arbconst(4) := ------- 8 ans; 2 2 - 8*log(w) - 8*log(z) - 4*w + 4*z - 15 {-------------------------------------------=0} 8 clear arbconst(!!arbconst) $ nodepend z,w $ % Now a homogeneous one depend y,x $ ode:=df(y,x) - (x-y)/(x+y); df(y,x)*x + df(y,x)*y - x + y ode := ------------------------------- x + y % To make this look decent... for all x,w let e**((log x)/w)=x**(1/w), (sqrt w)*(sqrt x)=sqrt(w*x) $ ans := odesolve(ode,y,x); This is a first-order ODE of algebraically homogeneous type solved by change of variables y = vx method 2 2 ans := {arbconst(5) + sqrt( - x + 2*x*y + y )=0} % Reducible to homogeneous % Note this is the previous example with origin shifted ode:=df(y,x) - (x-y-3)/(x+y-1); df(y,x)*x + df(y,x)*y - df(y,x) - x + y + 3 ode := --------------------------------------------- x + y - 1 ans := odesolve(ode,y,x); This is a first-order ODE reducible to homogeneous type solved by shifting the origin 2 2 ans := {arbconst(6) + sqrt( - x + 2*x*y + 6*x + y - 2*y - 7)=0} % and the special case of reducible to homogeneous ode:=df(y,x)-(2*x+3*y+1)/(4*x+6*y+1); 4*df(y,x)*x + 6*df(y,x)*y + df(y,x) - 2*x - 3*y - 1 ode := ----------------------------------------------------- 4*x + 6*y + 1 ans := odesolve(ode,y,x); This is a first-order ODE reducible to homogeneous type belonging to the special case where top and bottomare parallel lines solved by new variable and separation 49*arbconst(7) - 3*log(14*x + 21*y + 5) - 21*x + 42*y ans := {-------------------------------------------------------=0} 49 % To tidy up the next one we need for all x,w let e**(log x + w) = x*e**w, e**(w*log x)=x**w $ % a Bernoulli equation ode:=x*(1-x**2)*df(y,x) + (2*x**2 -1)*y - x**3*y**3; 3 3 3 2 ode := - df(y,x)*x + df(y,x)*x - x *y + 2*x *y - y odesolve(ode,y,x); This is a first-order ODE of Bernoulli type 5 1 5*arbconst(8) + 2*x {----=----------------------} 2 4 2 y 5*x - 5*x % and finally, in this set, an exact case ode:=(2*x**3 - 6*x*y + 6*x*y**2) + (-3*x**2 + 6*x**2*y - y**3)*df(y,x); 2 2 3 3 2 ode := 6*df(y,x)*x *y - 3*df(y,x)*x - df(y,x)*y + 2*x + 6*x*y - 6*x*y odesolve(ode,y,x); This is an exact first order ODE 4 2 2 2 4 {4*arbconst(9) + 2*x + 12*x *y - 12*x *y - y =0} % ****************************************** % Now for higher-order linear equations with constant coefficients % First, examples without driving terms % A simple one to start ode:=6*df(y,x,2)+df(y,x)-2*y; ode := 6*df(y,x,2) + df(y,x) - 2*y odesolve(ode,y,x); This is a linear ODE with constant coefficients of order 2 (7*x)/6 arbconst(11) + e *arbconst(10) {y=--------------------------------------} (2*x)/3 e % An example with repeated and complex roots ode:=df(y,x,4)+2*df(y,x,2)+y; ode := df(y,x,4) + 2*df(y,x,2) + y odesolve(ode,y,x); This is a linear ODE with constant coefficients of order 4 {y= - arbconst(15)*sin(x)*x + arbconst(14)*cos(x)*x - arbconst(13)*sin(x) + arbconst(12)*cos(x)} % A simple right-hand-side using the above example; % It will need the substitution for all w let (sin w)**2 + (cos w)** 2 = 1 $ ode:=ode-exp(x); x ode := df(y,x,4) + 2*df(y,x,2) - e + y odesolve(ode,y,x); This is a linear ODE with constant coefficients of order 4 {y=( - 4*arbconst(19)*sin(x)*x + 4*arbconst(18)*cos(x)*x - 4*arbconst(17)*sin(x) x + 4*arbconst(16)*cos(x) + e )/4} ode:=df(y,x,2)+4*df(y,x)+4*y-x*exp(x); x ode := df(y,x,2) + 4*df(y,x) - e *x + 4*y ans:=odesolve(ode,y,x); This is a linear ODE with constant coefficients of order 2 3*x 3*x 27*arbconst(21)*x + 27*arbconst(20) + 3*e *x - 2*e ans := {y=---------------------------------------------------------} 2*x 27*e % At x=1 let y=0 and df(y,x)=1 ans2 := solve({first ans, 1 = df(rhs first ans, x)}, {arbconst(!!arbconst-1),arbconst(!!arbconst)}); 2*x x 2 x x e *(9*e *x - 6*e *x + 2*e - 54*x*y - 27*x + 27*y) ans2 := {{arbconst(20)=-------------------------------------------------------, 27 2*x x x e *( - 3*e *x + e + 18*y + 9) arbconst(21)=----------------------------------}} 9 arbconst(!!arbconst -1) := sub(x=1,y=0,rhs first first ans2); 2 e *(5*e - 27) arbconst(20) := --------------- 27 arbconst(!!arbconst) := sub(x=1,y=0,rhs second first ans2); 2 e *( - 2*e + 9) arbconst(21) := ----------------- 9 ans; 3*x 3*x 3 3 2 2 3*e *x - 2*e - 6*e *x + 5*e + 27*e *x - 27*e {y=-----------------------------------------------------} 2*x 27*e clear arbconst(!!arbconst),arbconst(!!arbconst-1), ans, ans2 $ % For simultaneous equations you can use the machine e.g. as follows depend z,x $ ode1:=df(y,x,2)+5*y-4*z+36*cos(7*x); ode1 := 36*cos(7*x) + df(y,x,2) + 5*y - 4*z ode2:=y+df(z,x,2)-99*cos(7*x); ode2 := - 99*cos(7*x) + df(z,x,2) + y ode:=df(ode1,x,2)+4*ode2; ode := - 2160*cos(7*x) + df(y,x,4) + 5*df(y,x,2) + 4*y y := rhs first odesolve(ode,y,x); This is a linear ODE with constant coefficients of order 4 y := arbconst(25)*sin(x) + arbconst(24)*cos(x) - arbconst(23)*sin(2*x) + arbconst(22)*cos(2*x) + cos(7*x) z := rhs first solve(ode1,z); z := (4*arbconst(25)*sin(x) + 4*arbconst(24)*cos(x) - arbconst(23)*sin(2*x) + arbconst(22)*cos(2*x) - 8*cos(7*x))/4 clear ode1, ode2, ode, y,z $ nodepend z,x $ % A "homogeneous" n-th order (Euler) equation ode := x*df(y,x,2) + df(y, x) + y/x + (log x)**3; 2 3 df(y,x,2)*x + df(y,x)*x + log(x) *x + y ode := ------------------------------------------ x odesolve(ode, y, x); This equation is of the homogeneous (Euler) type 3 {y=( - 2*arbconst(27)*sin(log(x)) + 2*arbconst(26)*cos(log(x)) - log(x) *x 2 + 3*log(x) *x - 3*log(x)*x)/2} % Not yet working % ode :=6*df(y,x,2)+df(y,x)-2*y + tan x; % odesolve(ode, y,x); % To reset the system !!arbconst := 0 $ clear ode $ off trode$ nodepend y,x $ end $ (TIME: odesolve 2950 3310)