Nonhomogeneous Differential Equations — Question 8

Question 8 – Solution

Strategy. Recompute the Wronskian identity with the forcing included, then distinguish linear independence from a homogeneous fundamental set.

Step 1: Keep the forcing terms. Differentiating and substituting gives $$W'=uv''-u''v=u(g-pv'-qv)-v(g-pu'-qu),$$ so $$\boxed{W'=-pW+g(u-v).}$$ The extra term is absent from Abel’s homogeneous identity and generally does not vanish.

Step 2: Verify the example. We have $u''+u=\cos t+1-\cos t=1$, and adding $\sin t$ preserves this forcing. Direct computation gives $$\boxed{W=(1-\cos t)\cos t-\sin^2t=\cos t-1.}$$ It is zero at $0$ but equals $-2$ at $\pi$. Its derivative is $-\sin t$, matching $g(u-v)=-\sin t$. The homogeneous zero-or-never-zero conclusion is inapplicable to this forced pair.

Step 3: Test independence. If $Au+Bv=0$, evaluation at $\pi$ gives $2A+2B=0$ and at $\pi/2$ gives $A+2B=0$. Hence $A=B=0$. The difference $v-u=\sin t$ solves the associated homogeneous equation. Independence of the forced functions does not make them a fundamental set of homogeneous solutions.

Step 4: Describe their affine family. The forcing of $Au+Bv$ is $A+B$, so retaining forcing $1$ requires $A+B=1$. Such combinations are $$\boxed{u+B\sin t=1-\cos t+B\sin t.}$$ They have initial value zero and arbitrary slope $B$. They give exactly the forced solutions with initial value zero, but miss every solution with a different initial value. The full family is $1+C\cos t+D\sin t$, with two independent initial-data freedoms.