3 Systems of differential equations Example 7. Let V be the set of all vector-valued solutions x(t)= of the vector differential equation (2) x=Ax, V is a vector space under the usual Operations of vector addition and scalar multiplication. To wit, observe that axioms (i), (ii), and (v)-(viii) are automatically satisfied. Hence, we need only verify that (a) The sum of any two solutions of (2) is again a solution. (b) A constant times a solution of (2) is again a solution. (c) The vector-valued function x(t)= x 1 ~t) =[~] xn(t) 0 is a solution of (2) (axiom (iii)). (d) The negative of any solution of (2) is again a solution (axiom (iv)). Now (a) and (b) are exactly Theorem I of the previous section, while (d) is a special case of (b). To verify (c) we observe that Hence the vector-valued function x(t)=:O is always a solution of the differential equation (2). ExERCISEs In each of Problems 1-6, determine whether the given set of elements form a vector space under the properties of vector addition and scalar multiplication defined in Section 3.1. 278 3.3 Dimension of a vector space 1. The set of all elements x= ( ~~) where 3x -2x =0 2. The set of all elements x = ( :~ ) where x + x + x = 0 3. The set of all elements x = ( :~ ) where xf + xi + xj = 1 4. The set of all elements x = ( ~~) where x 2 1 1 3 2 1 + x2 + x3 = 1 5. The set of elements x = ( %) for all real numbers a and b 6. The set of all elements x = ( x 1 +x2 +x3 =0, :i) where x 1 -x2 +2x3 =0, 3x 1 -x2 +5x3 =0 In each of Problems 7-11 determine whether the given set of functions form a vector space under the usual operations of function addition and multiplication of a function by a constant. 7. The set of all polynomials of degree .;;; 4 8. The set of all differentiable functions 9. The set of all differentiable functions whose derivative at t = 1 is three 10. The set of all solutions of the differential equationy"+y=cost 11. The set of all functionsy(t) which have period 2'll', that isy(t+2'll')=y(t) 12. Show that the set of all vector-valued solutions x(t)=(x 1 (t)) x 2 (t) of the system of differential equations dx 1 - =x2 + 1 ' dt is not a vector space. 3.3 Dimension of a vector space Let V be the set of all solutions y(t) of the second-order linear homogeneous equation (d 2y I dt 2 ) + p(t)(dy I dt) + q(t)y =0. Recall that every solutiony(t) can be expressedas a linear combination of any two linearly independent solutions. Thus, if we knew two "independent" functions y 1( t) and 279