Solve Differential Equation
Solve a differential equation analytically by using the
function, with or without initial conditions. To solve a system of differential equations, see
Solve a System of Differential Equations.
First-Order Linear ODE
Solve this differential equation.
First, represent y by using
syms to create the
Define the equation using
== and represent differentiation using the
ode = diff(y,t) == t*y
ode(t) = diff(y(t), t) == t*y(t)
Solve the equation using
ySol(t) = dsolve(ode)
ySol(t) = C1*exp(t^2/2)
Solve Differential Equation with Condition
In the previous solution, the constant
C1 appears because no condition
was specified. Solve the equation with the initial condition
y(0) == 2. The
dsolve function finds a value of
C1 that satisfies
cond = y(0) == 2; ySol(t) = dsolve(ode,cond)
ySol(t) = 2*exp(t^2/2)
dsolve cannot solve your
equation, then try solving the equation numerically. See Solve a Second-Order Differential Equation Numerically.
Nonlinear Differential Equation with Initial Condition
Solve this nonlinear differential equation with an initial condition. The equation has multiple solutions.
syms y(t) ode = (diff(y,t)+y)^2 == 1; cond = y(0) == 0; ySol(t) = dsolve(ode,cond)
ySol(t) = exp(-t) - 1 1 - exp(-t)
Second-Order ODE with Initial Conditions
Solve this second-order differential equation with two initial conditions.
Define the equation and conditions. The second initial condition involves the first
y. Represent the derivative by creating the symbolic function
Dy = diff(y) and then define the condition using
syms y(x) Dy = diff(y); ode = diff(y,x,2) == cos(2*x)-y; cond1 = y(0) == 1; cond2 = Dy(0) == 0;
y. Simplify the solution using the
conds = [cond1 cond2]; ySol(x) = dsolve(ode,conds); ySol = simplify(ySol)
ySol(x) = 1 - (8*sin(x/2)^4)/3
Third-Order ODE with Initial Conditions
Solve this third-order differential equation with three initial conditions.
Because the initial conditions contain the first- and second-order derivatives, create two
Du = diff(u,x) and
diff(u,x,2), to specify the initial conditions.
syms u(x) Du = diff(u,x); D2u = diff(u,x,2);
Create the equation and initial conditions, and solve it.
ode = diff(u,x,3) == u; cond1 = u(0) == 1; cond2 = Du(0) == -1; cond3 = D2u(0) == pi; conds = [cond1 cond2 cond3]; uSol(x) = dsolve(ode,conds)
uSol(x) = (pi*exp(x))/3 - exp(-x/2)*cos((3^(1/2)*x)/2)*(pi/3 - 1) -... (3^(1/2)*exp(-x/2)*sin((3^(1/2)*x)/2)*(pi + 1))/3
More ODE Examples
This table shows examples of differential equations and their Symbolic Math Toolbox™ syntax. The last example is the Airy differential equation, whose solution is called the Airy function.
syms y(t) ode = diff(y)+4*y == exp(-t); cond = y(0) == 1; ySol(t) = dsolve(ode,cond)
ySol(t) = exp(-t)/3 + (2*exp(-4*t))/3
syms y(x) ode = 2*x^2*diff(y,x,2)+3*x*diff(y,x)-y == 0; ySol(x) = dsolve(ode)
ySol(x) = C2/(3*x) + C3*x^(1/2)
The Airy equation.
syms y(x) ode = diff(y,x,2) == x*y; ySol(x) = dsolve(ode)
ySol(x) = C1*airy(0,x) + C2*airy(2,x)