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% Miscellaneous ODESolve 1+ tests % Check for a problem in 1.03, spotted by David Hartley % <DHartley@physics.adelaide.edu.au>, caused by the reval in % `get_k_list' with caching enabled. The following should all give % the same result: odesolve(df(u,x,x)=df(u,x)); odesolve(df(u,x,2)=df(u,x)); odesolve(df(u,x,x)=df(u,x), u, x); % Linear first-order ODE: odesolve(df(y,t) = -w*y*tan(w*t - d)); % The solution, by inspection, is y = A cos(w t - d) % Variation of parameters: depend y, x; ode := df(y,x,2) + y - csc(x)$ odesolve(ode, y, x); sub(ws, ode); trigsimp ws; ode := 2*df(y,x,2) + y - csc(x)$ odesolve(ode, y, x); sub(ws, ode); trigsimp ws; % Bernoulli: ode := df(y,x)*y*x^2 - y^2*x - x^3 + 1; odesolve(ode, y, x, explicit); sub(ws, ode); % Implicit dependence: % (NB: Wierd constants need to be mopped up by the arbconst % simplification code!) % These should all behave equivalently: operator f, g; depend {y, ff}, x, {gg}, y; odesolve(df(y,x) = f(x), y, x); odesolve(df(y,x) = ff, y, x); odesolve(df(y,x) = g(y), y, x); odesolve(df(y,x) = gg, y, x); odesolve(df(y,x) = f(x)*g(y), y, x); odesolve(df(y,x) = ff*gg, y, x); odesolve(df(y,x) = 1/f(x)*g(y), y, x); odesolve(df(y,x) = 1/ff*gg, y, x); odesolve(df(y,x) = f(x)/g(y), y, x); odesolve(df(y,x) = ff/gg, y, x); % These should all fail (they are too implicit): depend {ff}, y, {gg}, x; odesolve(df(y,x) = ff, y, x); odesolve(df(y,x) = gg, y, x); odesolve(df(y,x) = ff*gg, y, x); odesolve(df(y,x) = 1/ff*gg, y, x); odesolve(df(y,x) = ff/gg, y, x); % NONlinear ODEs: odesolve(df(y,x) + y**(5/3)*arbconst(-1)=0); % Do not re-evaluate the solution without turning the algint switch on! odesolve(df(y,x,2) + c/(y^2 + k^2)^(3/2) = 0, y, x, algint); % Good test of ODESolve!-Alg!-Solve. Takes forever with fullroots on, % but with fullroots off ODESolve solves it. (Slightly tidier with % algint, but not necessary. However, the explicit option misses the % non-trivial solution that can fairly easily be found by hand!) odesolve(df(y,x,3) = 6*df(y,x)*df(y,x,2)/y - 6*df(y,x)^3/(y^2), y, x, algint); % Hangs with algint option! % off odesolve_plus_or_minus; odesolve(a*tan(asin((df(y,x) - y)/(2*y))/2)^2 + a - 2*sqrt(3)*tan(asin((df(y,x) - y)/(2*y))/2)*y + 4*sqrt(3)*y + tan(asin((df(y,x) - y)/(2*y))/2)^2*y - 4*tan(asin((df(y,x) - y)/(2*y))/2)*y + 7*y, y, x); % on odesolve_plus_or_minus; % From: K Sudhakar <ks@maths.qmw.ac.uk> odesolve(2*df(f,x,3)*df(f,x)*f^2*x^2 - 3*df(f,x,2)^2*x^2*f^2 + df(f,x)^4*x^2 - df(f,x)^2*f^2, f, x); % Related intermediate problem: odesolve(2*df(y,x)*x*y + x^2 - 2*x*y - y^2, y, x, explicit); % Anharmonic oscillator problem (which apparently Maple V R5.1 solves % in terms of a root of an expression involving unevaluated integrals % but Maple 6 cannot!). % General solution: odesolve(M*L*df(phi(tt),tt,2) = -M*g*sin(phi(tt))); % Use of `t' as independent variable: odesolve(M*L*df(phi(t),t,2) = -M*g*sin(phi(t))); % Conditional (eigenvalue) solution: %% odesolve(M*L*df(phi(t),t,2) = -M*g*sin(phi(t)), %% {t=0, phi(t)=0, df(phi(t),t)=Pi}); %% %% Conditional solutions need more work! This fails with %% ***** 0 invalid as kernel % Try setting %% L:=1; g:=10; ws; end;