Show that the solution of the initial value problem y ?? + p ( t ) y ? + q ( t )

Question

Show that the solution of the initial value problem
y?? + p(t)y? + q(t)y = g(t), y(t0) = y0, y?(t0) = y0?

can be written as y = u(t) + v(t), where u and v are solutions of the two initial value

problems

u?? + p(t)u? + q(t)u = 0, u(t0) = y0, u?(t0) = y0? , v?? + p(t)v? + q(t)v = g(t), v(t0) = 0, v?(t0) = 0,

respectively. In other words, the nonhomogeneities in the differential equation and in the initial conditions can be dealt with separately. (Note that u is easy to find if a fundamental set of solutions of u?? + p(t)u? + q(t)u = 0 is known.)

Explanation / Answer

L[y] = y'' + p(t)y' + q(t)y = g(t); y(t0) = y0; y0(t0) = y'0 -------------- (i)

can be written as y = u(t) + v(t), where u and v are solutions of two initial value problems

L[u] = 0; u(t0) = y0; u'(t0) = y0 ------------------ (ii)

L[v] = g(t); v(t0) = 0; v'(t0) = 0

Suppose y is a solution of equation (i), u is a solution of equation (ii), Let v = y - u,

then

L[y] = y'' + p(t)y' + q(t)y = g(t); y(t0) = y0; y'(t0) = y0

L[u] = u'' + p(t)u' + q(t)u = 0; u(t0) = y0; u'(t0) = y0

Hence

L[v] = L[y - u] = g(t) and v(t0) = 0; v0(t0) = 0:

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.