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Solve a differential equation analytically by using the dsolve function, with or without initial conditions. To solve a system of differential equations, see Solve a System of Differential Equations.
Solve this differential equation.
First, represent y by using syms to create the symbolic function y(t) .
syms y(t)
Define the equation using == and represent differentiation using the diff function.
ode = diff(y,t) == t*y
ode(t) = diff(y(t), t) == t*y(t)
Solve the equation using dsolve .
ySol(t) = dsolve(ode)
ySol(t) = C1*exp(t^2/2)
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 the condition.
cond = y(0) == 2; ySol(t) = dsolve(ode,cond)
ySol(t) = 2*exp(t^2/2)
If dsolve cannot solve your equation, then try solving the equation numerically. See Solve a Second-Order Differential Equation Numerically.
Solve this nonlinear differential equation with an initial condition. The equation has multiple solutions.
( d y d t + y ) 2 = 1 , y ( 0 ) = 0.
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)
Solve this second-order differential equation with two initial conditions.
d 2 y d x 2 = cos ( 2 x ) − y , y ( 0 ) = 1 , y ' ( 0 ) = 0.
Define the equation and conditions. The second initial condition involves the first derivative of y . Represent the derivative by creating the symbolic function Dy = diff(y) and then define the condition using Dy(0)==0 .
syms y(x) Dy = diff(y); ode = diff(y,x,2) == cos(2*x)-y; cond1 = y(0) == 1; cond2 = Dy(0) == 0;
Solve ode for y . Simplify the solution using the simplify function.
conds = [cond1 cond2]; ySol(x) = dsolve(ode,conds); ySol = simplify(ySol)
ySol(x) = 1 - (8*sin(x/2)^4)/3
Solve this third-order differential equation with three initial conditions.
d 3 u d x 3 = u , u ( 0 ) = 1 , u ′ ( 0 ) = − 1 , u ′ ′ ( 0 ) = π .
Because the initial conditions contain the first- and second-order derivatives, create two symbolic functions, Du = diff(u,x) and D2u = 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
This table shows examples of differential equations and their Symbolic Math Toolbox™ syntax.
d y d t + 4 y ( t ) = e − t , y ( 0 ) = 1.
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
2 x 2 d 2 y d x 2 + 3 x d y d x − y = 0.
syms y(x) ode = 2*x^2*diff(y,x,2)+3*x*diff(y,x)-y == 0; ySol(x) = dsolve(ode)
ySol(x) = C1/(3*x) + C2*x^(1/2)
The Airy equation.
d 2 y d x 2 = x y ( x ) .
syms y(x) ode = diff(y,x,2) == x*y; ySol(x) = dsolve(ode)
ySol(x) = C1*airy(0,x) + C2*airy(2,x)
Puiseux series solution.
( x 2 + 1 ) d 2 y d x 2 − 2 x d y d x + y = 0.
syms y(x) a ode = (x^2+1)*diff(y,x,2)-2*x*diff(y,x)+y == 0; Dy = diff(y,x); cond = [Dy(0) == a; y(0) == 5]; ySol(x) = dsolve(ode,cond,'ExpansionPoint',0)
ySol(x) = - (a*x^5)/120 - (5*x^4)/24 + (a*x^3)/6 - (5*x^2)/2 + a*x + 5
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