Differential Equations

Brown University Applied Mathematics

MuPAD Differential Equation Solvers

When you are typing up your homework, it can be helpful to check your work. MuPAD has several solvers that you can use to check your work and that may save you time when there is no need to show your work. The different solvers that MuPAD has are:

Examples

Ode() Solver

eq1 := ode(y''(x)-3*y'(x)+6*y(x), y(x))

ode((D@@2)(y)(x) - 3*D(y)(x) + 6*y(x), y(x))

solve(%)

{C2*exp((3*x)/2)*cos((15^(1/2)*x)/2) + C3*exp((3*x)/2)*sin((15^(1/2)*x)/2)}

 

Wronskian Solver

eq2 := ode::wronskian([3*x^2+1, x*sqrt(1+x^2)], x)

(3*x^2 + 1)*((x^2 + 1)^(1/2) + x^2/(x^2 + 1)^(1/2)) - 6*x^2*(x^2 + 1)^(1/2)

eq3 := ode::wronskian([3*x^5, 9*x^5], x)

0

 

Special Ode Solver

eq4 := ode::solve(y'(x) = 1/x*y(x)^2 + 1/x*y(x) + x, y(x), Type = Chini)

{-x*I, x*tan(C8 + x), x*I}

eq5 := ode::solve(y'(x) = (- 1/x + 2*I)*y(x) + 1/x*y(x)^2, y(x), Type = Bernoulli)

{0, (x*exp(- ln(x) + 2*x*I))/(exp(2*x*I) + C11*x + 2*x*Ei(1, -2*x*I)*I)}

 

Numeric Solutions - numeric::odesolver

IVP := {y''(t) = t*y'(t), y(0) = 0, y'(0) = 2}

{D(y)(0) = 2, y(0) = 0, (D@@2)(y)(t) = t*D(y)(t)}

fields := [y(t), y'(t)]

[y(t), D(y)(t)]

[ODE, t0, Y0] := [numeric::ode2vectorfield(IVP, fields)]

[`proc ODE(t, Y) ... end`, 0, [0, 2]]

numeric::odesolve(ODE, t0..3, Y0)

[70.78350917, 180.0342626]

 

Numeric Solutions - numeric::odesolver2

IVP := {y''(t) = t*y'(t), y(0) = 0, y'(0) = 12}

{D(y)(0) = 12, y(0) = 0, (D@@2)(y)(t) = t*D(y)(t)}

fields := [y(t), y'(t)]

[y(t), D(y)(t)]

ODE := numeric::ode2vectorfield(IVP, fields)

`proc(t, Y) ... end`, 0, [0, 12]

numApprox := numeric::odesolve2(ODE)

`proc numApprox(t) ... end`

numApprox(0);

[0.0, 12.0]

numApprox(4);

[9672.479972, 35771.49584]

numApprox(5/17)

[3.580964306, 12.53041946]

plotfunc2d(numApprox(t)[1], t = 0..4)

MuPAD graphics

 

Linear Ordinary Differential Equation Tester

ode::isLODE(y(x)^2+x^3*diff(y(x),x)-x, y(x))

FALSE

ode::isLODE(y''(x)+16*y'(x)-16, y(x))

TRUE


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