- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
2.4 Exercises
In each of Exercises 24 through 27, a transformation $T: V \rightarrow V$ is described as indicated. In each case, determine whether $T$ is linear. If $T$ is linear, describe its null space and range, and compute the nullity and rank when they are finite.
27. $\quad$ Let $V$ be the space of all real functions twice differentiable on an open interval $(a,b).$ If $y\in V,$ define $T(y)=y'' + P\,y' + Q\,y,$ where $P$ and $Q$ are fixed constants.
Solution. $\quad$ Recall from Section 2.1 that if $V$ and $W$ are linear spaces, then a function $T: V \to W$ is called a linear transformation from $V$ into $W$ if it has the following two properties:
$\quad$ (a) $\quad T(x + y) = T(x) + T(y) \quad$ for all $x$ and $y$ in $V,$
$\quad$ (b) $\quad T(cx) = cT(x) \quad$ for all $x$ in $V$ and all scalars $c.$
$\quad$ Let $x$ and $y$ be two real functions twice differentiable on the open interval $(a, b).$ By the additivity and scalar muliplication of the derivative, we know that $(x + y)' = x' + y'$ and $(cx)' = cx',$ giving us: \begin{align*} T(x + y) &= (x + y)'' + P(x + y)' + Q(x + y) \\ &= x'' + Px' + Qx + y'' + Py' + Qy \\ &= T(x) + T(y) \\ \\ T(cx) &= (cx)'' + P(cx)' + Q(cx) \\ &= cx'' + cPx' + cQx \\ &= cT(x) \end{align*} From this we can see that $T$ is a linear transformation.
$\quad$ The range of $T$ is the set of all second-order linear differential equations $f(y) = y'' + Py' + Qy,$ defined on the open interval $(a, b).$ As demonstrated in Exercise 25, the rank of $T$ is infinite.
$\quad$ The null space of $T$ is the set of functions $y$ whose second-order differential equations with constant coefficients are equal to zero. From Theorem 8.7 of Volume 1, we recall that the solutions to this equation depend on the discriminant of the equation:
$\quad$Theorem 8.7. $\quad$ Let $d = P^2 - 4Q$ be the discriminant of the linear differential equation $y'' + Py' + Qy = 0.$ Then, every solution of this equation on $(a, b)$ has the form:
\begin{align*}
y = e^{-Px/2}\left[c_1u_1(t) + c_2u_2(t)\right],
\end{align*}
where $c_1$ and $c_2$ are constants, and the functions $u_1$ and $u_2$ are determined according to the algebraic sign of the discriminant as follows:
$\quad$ (a) If $d = 0,$ then $u_1(t) = 1$ and $u_2(t) = t.$
$\quad$ (b) If $d \gt 0,$ then $u_1(t) = e^{kx}$ and $u_2(t) = e^{-kx},$ where $k = \frac{1}{2}\sqrt{d}.$
$\quad$ (b) If $d \lt 0,$ then $u_1(t) = \cos kx$ and $u_2(t) = \sin kx,$ where $k = \frac{1}{2}\sqrt{-d}.$
Regardless of the sign of the discriminant, the nullity of $T$ is $2. \quad \blacksquare$