DSOLVE Symbolic solution of ordinary differential equations.
DSOLVE will not accept equations as strings in a future release.
Use symbolic expressions or sym objects instead.
For example, use syms y(t); dsolve(diff(y)==y) instead of dsolve('Dy=y').
DSOLVE(eqn1,eqn2, ...) accepts symbolic equations representing
ordinary differential equations and initial conditions.
By default, the independent variable is 't'. The independent variable
may be changed from 't' to some other symbolic variable by including
that variable as the last input argument.
The DIFF function constructs derivatives of symbolic functions (see sym/symfun).
Initial conditions involving derivatives must use an intermediate
variable. For example,
syms x(t)
Dx = diff(x);
dsolve(diff(Dx) == -x, Dx(0) == 1)
If the number of initial conditions given is less than the
number of dependent variables, the resulting solutions will obtain
arbitrary constants, C1, C2, etc.
Three different types of output are possible. For one equation and one
output, the resulting solution is returned, with multiple solutions to
a nonlinear equation in a symbolic vector. For several equations and
an equal number of outputs, the results are sorted in lexicographic
order and assigned to the outputs. For several equations and a single
output, a structure containing the solutions is returned.
If no closed-form (explicit) solution is found, then a
warning is given and the empty sym is returned.
DSOLVE(...,'IgnoreAnalyticConstraints',VAL) controls the level of
mathematical rigor to use on the analytical constraints of the solution
(branch cuts, division by zero, etc). The options for VAL are TRUE or
FALSE. Specify FALSE to use the highest level of mathematical rigor
in finding any solutions. The default is TRUE.
DSOLVE(...,'MaxDegree',n) controls the maximum degree of polynomials
for which explicit formulas will be used in SOLVE calls during the
computation. n must be a positive integer smaller than 5.
The default is 2.
DSOLVE(...,'Implicit',true) returns the solution as a vector of
equations, relating the dependent and the independent variable. This
option is not allowed for systems of differential equations.
DSOLVE(...,'ExpansionPoint',a) returns the solution as a series around
the expansion point a.
DSOLVE(...,'Order',n) returns the solution as a series with order n-1.
Examples:
% Example 1
syms x(t) a
dsolve(diff(x) == -a*x) returns
ans = C1/exp(a*t)
% Example 2: changing the independent variable
x = dsolve(diff(x) == -a*x, x(0) == 1, 's') returns
x = 1/exp(a*s)
syms x(s) a
x = dsolve(diff(x) == -a*x, x(0) == 1) returns
x = 1/exp(a*s)
% Example 3: solving systems of ODEs
syms f(t) g(t)
S = dsolve(diff(f) == f + g, diff(g) == -f + g,f(0) == 1,g(0) == 2)
returns a structure S with fields
S.f = (i + 1/2)/exp(t*(i - 1)) - exp(t*(i + 1))*(i - 1/2)
S.g = exp(t*(i + 1))*(i/2 + 1) - (i/2 - 1)/exp(t*(i - 1))
syms f(t) g(t)
v = [f;g];
A = [1 1; -1 1];
S = dsolve(diff(v) == A*v, v(0) == [1;2])
returns a structure S with fields
S.f = exp(t)*cos(t) + 2*exp(t)*sin(t)
S.g = 2*exp(t)*cos(t) - exp(t)*sin(t)
% Example 3: using options
syms y(t)
dsolve(sqrt(diff(y))==y) returns
ans = 0
syms y(t)
dsolve(sqrt(diff(y))==y, 'IgnoreAnalyticConstraints', false) warns
Warning: The solutions are subject to the following conditions:
(C67 + t)*(1/(C67 + t)^2)^(1/2) = -1
and returns
ans = -1/(C67 + t)
% Example 4: Higher order systems
syms y(t) a
Dy = diff(y);
D2y = diff(y,2);
dsolve(D2y == -a^2*y, y(0) == 1, Dy(pi/a) == 0)
syms w(t)
Dw = diff(w);
D2w = diff(w,2);
w = dsolve(diff(D2w) == -w, w(0)==1, Dw(0)==0, D2w(0)==0)
See also SOLVE, SUBS, SYM/DIFF, odeToVectorfield.
Documentation for dsolve
doc dsolve
