# Tema 12
12-1  Familias de curvas planas
          Ecuación diferencial de una familia de curvas

## Familias de curvas

kill(all)$

Ejemplo 12.1.1

f11(x):= 0.5*x$           f12(x):=x$        f13(x):=3*x$        
f14(x):=-0.5*x$           f15(x):=-x$       f16(x):=-3*x$

wxplot2d([f11(x),f12(x),f13(x),f14(x),f15(x),f16(x)], [x,-2,2], 
[style, [lines, 2]] , [legend, ""],
 [gnuplot_preamble, "set size ratio 1; set grid;"] , [nticks, 300]  )$

Ejemplo 12.1.2

kill(all)$

f31(x):=x^2$           f32(x):=2*x^2$       f33(x):=3*x^2$        
f34(x):=-x^2$          f35(x):=-2*x^2$        f36(x):=-3*x^2$

wxplot2d([f31(x),f32(x),f33(x),f34(x),f35(x),f36(x)], [x,-2,2], 
 [style, [lines, 2]] , [legend, ""],
 [gnuplot_preamble, "set size ratio 1; set grid;"] , [nticks, 300]   )$

Ejemplo 12.1.3

kill(all)$

f11(x):=exp(0.5*x)$     f12(x):=exp(x)$
f13(x):=exp(1.5*x)$     f14(x):=exp(2*x)$
f21(x):=exp(-0.5*x)$    f22(x):=exp(-x)$
f23(x):=exp(-1.5*x)$    f24(x):=exp(-2*x)$

wxplot2d([f11(x),f12(x),f13(x),f14(x),f21(x),f22(x),f23(x),f24(x)], [x,-1,1],
  [style, [lines, 2]] , [legend, ""],
 [gnuplot_preamble, "set size ratio 1; set grid;"] , [nticks, 300] )$

Ejemplo 12.1.4

kill(all)$

f11(x):=2 - exp(- 0.5*x)$     f12(x):=2 - exp(- 1.0*x)$
f13(x):=2 - exp(- 0.8*x)$     f14(x):=2 - exp(- 1.5*x)$

wxplot2d([f11(x),f12(x),f13(x),f14(x)], [x, 0, 8], [y, 1.0, 2.2], 
 [style, [lines, 2]] , [legend, ""],
 [gnuplot_preamble, "set size ratio 1;  set grid;"] , [nticks, 10] )$

Ejemplo 12.1.5

kill(all)$

f51(t):=cos(2*t) + sin(2*t)$     f52(t):=1.5*cos(2*t) + 1.5*sin(2*t)$
f53(t):=cos(2*t) - sin(2*t)$       f54(t):=1.5*cos(2*t) -  1.5*sin(2*t)$

wxplot2d([f51(t),f52(t),f53(t),f54(t)], [t,-4,4], 
 [style, [lines, 2]] , [legend, ""],
 [gnuplot_preamble, "set size ratio 1;  set grid;"] , [nticks, 10] )$

## Ecuación diferencial de una familia de curvas

### Familia uniparamétrica

kill(all)$

depends ([y] , [t]);

Eq1:  y=C*t^2;      
Eq2:  diff(Eq1,t) ;

eliminate([Eq1, Eq2],  [C]);

edo1: 2*y-t*'diff(y,t) = 0;

### Familia biparamétrica

kill(all)$

depends ([y] , [t]);

Eq1:  y = C1*cos(2*t) + C2*sin(2*t); 
Eq2:  diff(Eq1, t ) ;
Eq3:  diff(Eq2, t ) ;

eliminate([Eq1, Eq2, Eq3],  [C1,C2]);

edo2: 'diff(y,t,2)+4*y=0;

### Solución particular

Familia uniparamétrica

kill(all)$

y(t):=C*t^2;
Eq1: y(2)=5;

solve(Eq1, C);

C=C:5/4;

y=C*t^2;

Familia biparamétrica

kill(all)$

y(t):=C1*cos(2*t) + C2*sin(2*t);
define ( Dy(t) , diff(y(t),t) ) ;

Eq1:   y(0)=1;
Eq2:   Dy(0)=3;

C1=C1:1;        C2=C2:3/2;

'y(t) = y(t) ;