By Xu J.-J., Labute J.
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Dn ∈ R. Thus, if V is the solution space of the associated homogeneous DE L(y) = 0, the transformation T : V → Rn , defined by T (y) = (y(x0 ), y (x0 ), . . , y (n−1) (x0 )), is linear transformation of the vector space V into Rn since T (ay + bz) = aT (y) + bT (z). 52 FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS Moreover, the fundamental theorem says that T is one-to-one (T (y) = T (z) impliesy = z) and onto (every d ∈ Rn is of the form T (y) for some y ∈ V ). A linear transformation which is one-to-one and onto is called an isomorphism.
Here we again set z = y but try for a solution d d z as a function of y. Then, using the fact that dx = z dy , we get the DE z d dy n−1 (z) = f y, z, z dz d , . . , (z )n (z) dy dy which is of degree n − 1. For example, the DE y = (y )2 is of this type and we get the DE dz z = z2 dy which has the solution z = Cey . Hence y = Cey from which −e−y = Cx + D. This gives y = − log(−Cx − D) as the general solution which is in agreement with what we did previously. 4. 1 Linear Equations Basic Concepts and General Properties Let us now go to linear equations.
Heating and Cooling Problems Newton’s Law of Cooling states that the rate of change of the temperature of a cooling body is proportional to the difference between its temperature T and the temperature of its surrounding medium. Assuming the surroundings maintain a constant temperature Ts , we obtain the differential equation dT = −k(T − Ts ), dt where k is a constant. This is a linear DE with solution T = Ts + Ce−kt . If T (0) = T0 then C = T0 − Ts and T = Ts + (T0 − Ts )e−kt . As an example consider the problem of determining the time of death of a healthy person who died in his home some time before noon when his body was 70 degrees.