# Exercici 12.1

## Apartat a)

kill(all)$

define (y(t) , exp(-t) + t - 1);

y(0);

diff(y(t),t) + y(t) = t;

## Apartat b)

kill(all)$

define (y(t) , 8*cos(3*t) + 6*sin(3*t) );

define (Dy(t) , diff(y(t) , t) );

y(0);   Dy(0);

diff(y(t),t,2) + 9*y(t) = 0;

ratsimp(%);

## Apartat c)

kill(all)$

Eq1:  27*y(t)^2 - 8*t^3 = 0;

solve (Eq1 , y(t));

define (y(t) , (2^(3/2)*t^(3/2))/3^(3/2));

y(0);

define (Dy(t) , diff(y(t),t) );

(Dy(t))^3 - y(t);

## Apartat d)

kill(all)$

define (y(x) , A*x^3 + B*x^(-4) - x^2/3 );

define (Dy(x) , diff(y(x) , x) );
define (D2y(x) , diff(y(x) , x, 2) );

x^2*D2y(x) + 2*x*Dy(x) - 12*y(x) -2*x^2 ;

ratsimp(%);

## Apartat e)

kill(all)$

Eq1:  x*(y(x))^2 - (y(x))^3 - C = 0;

depends ([y] , [x]);
diff(Eq1 , x);

to_poly_solve([Eq2], [Dy(x)]);