MATH383

Overview

Constant-coefficient homogeneous equations
Non-homogeneous equations
Undetermined coefficients

Example 1. $y^{''} + 6 y^{'} + 5 y = 0, y (0) = 3, y^{'} (0) = - 1 .$

(%i3)

de: 'diff(y,x,2) + 6*'diff(y,x) + 5*y

$(%o3) \frac{d^{2} y}{d x^{2}} + 6 (\frac{d y}{d x}) + 5 y$

(%i4)

gsoln: ode2(de,y,x);

$(%o4) y = % k 1 e^{- x} + % k 2 e^{- 5 x}$

(%i6)

psoln: ic2(gsoln,x=0,y=3,'diff(y,x)=-1)

$(%o6) y = \frac{7 e^{- x}}{2} - \frac{e^{- 5 x}}{2}$

(%i11)

plot2d(rhs(psoln),[x,0,5],[ylabel,"y(x)"])$

Example 2. $y^{''} + 6 y^{'} + 9 y = 0, y (0) = 3, y^{'} (0) = - 1 .$

(%i12)

de: 'diff(y,x,2) + 6*'diff(y,x) + 9*y

$(%o12) \frac{d^{2} y}{d x^{2}} + 6 (\frac{d y}{d x}) + 9 y$

(%i13)

gsoln: ode2(de,y,x);

$(%o13) y = (% k 2 x + % k 1) e^{- 3 x}$

(%i14)

psoln: ic2(gsoln,x=0,y=3,'diff(y,x)=-1)

$(%o14) y = (8 x + 3) e^{- 3 x}$

(%i15)

plot2d(rhs(psoln),[x,0,5],[ylabel,"y(x)"])$

Example 3. $y^{''} + 4 y^{'} + 13 y = 0, y (0) = 2, y^{'} (0) = - 3 .$

(%i16)

de: 'diff(y,x,2) + 4*'diff(y,x) + 13*y

$(%o16) \frac{d^{2} y}{d x^{2}} + 4 (\frac{d y}{d x}) + 13 y$

(%i17)

gsoln: ode2(de,y,x);

$(%o17) y = e^{- 2 x} (% k 1 \sin (3 x) + % k 2 \cos (3 x))$

(%i18)

psoln: ic2(gsoln,x=0,y=2,'diff(y,x)=-3)

$(%o18) y = e^{- 2 x} (\frac{\sin (3 x)}{3} + 2 \cos (3 x))$

(%i19)

plot2d(rhs(psoln),[x,0,5],[ylabel,"y(x)"])$

Theorem. Consider $p, q, f$ continuous on $(a, b)$ , and $x_{0} \in (a, b)$ , $k_{0}, k_{1} \in ℝ$ . The initial value problem $y^{''} + p y^{'} + q y = f, y (x_{0}) = k_{0}, y^{'} (x_{0}) = k_{1},$ has a unique solution on $(a, b)$ .

Theorem. If ${y_{1}, y_{2}}$ is a fundamental solution set for $y^{''} + p y^{'} + q y = 0$ on $(a, b)$ , and $y_{p}$ is a particular solution of the $y^{''} + p y^{'} + q y = f$ on $(a, b)$ , then the solution of the problem $y^{''} + p y^{'} + q y = f, y (x_{0}) = k_{0}, y^{'} (x_{0}) = k_{1},$ is of the form

$y = y_{p} + c_{1} y_{1} + c_{2} y_{2} .$

Example 4. $y^{''} + y = 1, y (0), y^{'} (0) = 7$ . $y_{p} = 1$ , ${y_{1}, y_{2}} = {\cos x, \sin x}$

y = 1 + \cos x + 7 \sin x .

Example 5. $y^{''} - 2 y^{'} + y = - 3 - x + x^{2}, y (0) = - 2, y^{'} (0) = 1 .$

(%i24)

de: 'diff(y,x,2) - 2*'diff(y,x) + y + 3 + x -x^2

$(%o24) \frac{d^{2} y}{d x^{2}} - 2 (\frac{d y}{d x}) + y - x^{2} + x + 3$

(%i25)

gsoln: ode2(de,y,x);

$(%o25) y = (% k 1 + % k 2 x) e^{x} + x^{2} + 3 x + 1$

(%i26)

psoln: ic2(gsoln,x=0,y=-2,'diff(y,x)=1)

$(%o26) y = (x - 3) e^{x} + x^{2} + 3 x + 1$

(%i23)

plot2d(rhs(psoln),[x,0,5],[ylabel,"y(x)"])$

Example 6. $x^{2} y^{''} + x y^{'} - 4 y = 2 x^{4}, y (0) = - 2, y^{'} (0) = 1 .$

(%i27)

de: x^2*'diff(y,x,2) + x*'diff(y,x) -4*y - 2*x^2

$(%o27) x^{2} (\frac{d^{2} y}{d x^{2}}) + x (\frac{d y}{d x}) - 4 y - 2 x^{2}$

(%i28)

gsoln: ode2(de,y,x);

$(%o28) y = \frac{4 x^{2} \log (x) - x^{2}}{8} + % k 1 x^{2} + \frac{% k 2}{x^{2}}$

(%i30)

psoln: ic2(gsoln,x=1,y=1,'diff(y,x)=1)

$(%o30) y = \frac{4 x^{2} \log (x) - x^{2}}{8} + \frac{3 x^{2}}{4} + \frac{3}{8 x^{2}}$

(%i32)

plot2d(rhs(psoln),[x,1,5],[ylabel,"y(x)"])$

Consider equations of form $a y^{''} + b y^{'} + c y = e^{α x} G (x)$ , $a, b, c \in ℝ$ . The approach to finding a particular solution is to try the form $y_{p} = u e^{α x}$
1. For $G (x) = g$ , a constant, try $y_{p} = A e^{α x}$
  $A (a α^{2} + b α + c) = g, a α^{2} + b α + c \neq 0 \Rightarrow A = \frac{g}{a α^{2} + b α + c}$
2. If $a α^{2} + b α + c = 0$ , try $y_{p} = A x e^{α x}$
3. For general $G (x)$ , try $y_{p} = u (x) e^{α x}$

Example 7. $y^{''} - 3 y^{'} + 2 y = e^{3 x} (x^{2} + 2 x - 1), y (0) = 1, y^{'} (0) = 1 .$

(%i33)

de:'diff(y,x,2)-3*'diff(y,x)+2*y-exp(3*x)*(x^2+2*x-1)

$(%o33) \frac{d^{2} y}{d x^{2}} - 3 (\frac{d y}{d x}) + 2 y - (- 1 + 2 x + x^{2}) e^{3 x}$

(%i34)

gsoln: ode2(de,y,x);

$(%o34) y = \frac{(- 1 - 2 x + 2 x^{2}) e^{3 x}}{4} + % k 1 e^{2 x} + % k 2 e^{x}$

(%i36)

psoln: ic2(gsoln,x=0,y=1,'diff(y,x)=1)

$(%o36) y = \frac{(- 1 - 2 x + 2 x^{2}) e^{3 x}}{4} + e^{2 x} + \frac{e^{x}}{4}$

(%i38)

plot2d(rhs(psoln),[x,0,1],[ylabel,"y(x)"])$