Differential Equations | Part IA, 2002

The homogeneous equation

y¨+p(t)y˙+q(t)y=0\ddot{y}+p(t) \dot{y}+q(t) y=0

has non-constant, non-singular coefficients p(t)p(t) and q(t)q(t). Two solutions of the equation, y(t)=y1(t)y(t)=y_{1}(t) and y(t)=y2(t)y(t)=y_{2}(t), are given. The solutions are known to be such that the determinant

W(t)=y1y2y˙1y˙2W(t)=\left|\begin{array}{ll} y_{1} & y_{2} \\ \dot{y}_{1} & \dot{y}_{2} \end{array}\right|

is non-zero for all tt. Define what is meant by linear dependence, and show that the two given solutions are linearly independent. Show also that

W(t)exp(tp(s)ds).W(t) \propto \exp \left(-\int^{t} p(s) d s\right) .

In the corresponding inhomogeneous equation

y¨+p(t)y˙+q(t)y=f(t)\ddot{y}+p(t) \dot{y}+q(t) y=f(t)

the right-hand side f(t)f(t) is a prescribed forcing function. Construct a particular integral of this inhomogeneous equation in the form

y(t)=a1(t)y1(t)+a2(t)y2(t),y(t)=a_{1}(t) y_{1}(t)+a_{2}(t) y_{2}(t),

where the two functions ai(t)a_{i}(t) are to be determined such that

y1(t)a˙1(t)+y2(t)a˙2(t)=0y_{1}(t) \dot{a}_{1}(t)+y_{2}(t) \dot{a}_{2}(t)=0

for all tt. Express your result for the functions ai(t)a_{i}(t) in terms of integrals of the functions f(t)y1(t)/W(t)f(t) y_{1}(t) / W(t) and f(t)y2(t)/W(t)f(t) y_{2}(t) / W(t).

Consider the case in which p(t)=0p(t)=0 for all tt and q(t)q(t) is a positive constant, q=ω2q=\omega^{2} say, and in which the forcing f(t)=sin(ωt)f(t)=\sin (\omega t). Show that in this case y1(t)y_{1}(t) and y2(t)y_{2}(t) can be taken as cos(ωt)\cos (\omega t) and sin(ωt)\sin (\omega t) respectively. Evaluate f(t)y1(t)/W(t)f(t) y_{1}(t) / W(t) and f(t)y2(t)/W(t)f(t) y_{2}(t) / W(t) and show that, as tt \rightarrow \infty, one of the ai(t)a_{i}(t) increases in magnitude like a power of tt to be determined.

Typos? Please submit corrections to this page on GitHub.