### The Maxima on-line user's manual

Algebra Calculator

#### Ode2

Function: ode2 (<eqn>, <dvar>, <ivar>) The function `ode2` solves an ordinary differential equation (ODE) of first or second order. It takes three arguments: an ODE given by <eqn>, the dependent variable <dvar>, and the independent variable <ivar>. When successful, it returns either an explicit or implicit solution for the dependent variable. `%c` is used to represent the integration constant in the case of first-order equations, and `%k1` and `%k2` the constants for second-order equations. The dependence of the dependent variable on the independent variable does not have to be written explicitly, as in the case of `desolve`, but the independent variable must always be given as the third argument.

If `ode2` cannot obtain a solution for whatever reason, it returns `false`, after perhaps printing out an error message. The methods implemented for first order equations in the order in which they are tested are: linear, separable, exact - perhaps requiring an integrating factor, homogeneous, Bernoullis equation, and a generalized homogeneous method. The types of second-order equations which can be solved are: constant coefficients, exact, linear homogeneous with non-constant coefficients which can be transformed to constant coefficients, the Euler or equi-dimensional equation, equations solvable by the method of variation of parameters, and equations which are free of either the independent or of the dependent variable so that they can be reduced to two first order linear equations to be solved sequentially.

In the course of solving ODEs, several variables are set purely for informational purposes: `method` denotes the method of solution used (e.g., `linear`), `intfactor` denotes any integrating factor used, `odeindex` denotes the index for Bernoullis method or for the generalized homogeneous method, and `yp` denotes the particular solution for the variation of parameters technique.

In order to solve initial value problems (IVP) functions `ic1` and `ic2` are available for first and second order equations, and to solve second-order boundary value problems (BVP) the function `bc2` can be used.

Example:

```          (%i1) x^2*diff(y,x) + 3*y*x = sin(x)/x;
2 dy           sin(x)
(%o1)                x  -- + 3 x y = ------
dx             x
(%i2) ode2(%,y,x);
%c - cos(x)
(%o2)                    y = -----------
3
x
(%i3) ic1(%o2,x=%pi,y=0);
cos(x) + 1
(%o3)                   y = - ----------
3
x
(%i4) diff(y,x,2) + y*diff(y,x)^3 = 0;
2
d y      dy 3
(%o4)                   --- + y (--)  = 0
2      dx
dx
(%i5) ode2(%,y,x);
3
y  + 6 %k1 y
(%o5)                ------------ = x + %k2
6
(%i6) ratsimp(ic2(%o5,x=0,y=0,diff(y,x)=2));
3
2 y  - 3 y
(%o6)                   - ---------- = x
6
(%i7) bc2(%o5,x=0,y=1,x=1,y=3);
3
y  - 10 y       3
(%o7)                   --------- = x - -
6           2```

```(%o1)                                true
(%i2) ```

### Related Examples

ode2(%o1,D,x) ;

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##### ode2

Q: 'Diff(y(x),x)=(x-y...

Ans: ode2(Q,y(x),x) ;

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##### ode2-sin

eq1: dif(u,x) = sin(x...

funk: ode2(eq1,u,x);

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##### ode2

ode2(x^2*y-y,x,y);

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##### ode2-sin

diffeq:0=diff(r^2*dif...

ode2(diffeq, v-phi,[r...

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##### ode2-solve-tanh

fb: .1;

t1: y[n] = tanh(x[n] ...

solve(t1, y[n-1]);

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##### ode2

ode2((a-(b*t+3)^c*4)/...

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##### ode2-sin

eq1: dif(u,x) = sin(x...

funk: ode2(%,u,x);

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##### ode2

Q: 'Diff(y(x),x)=(-5*...

Ans: ode2(%,y,x) ;

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