Linear Homogeneous Differential Equations — Question 8

Question 8 – Solution

Strategy. Compare each exponent and its polynomial degree, distinguishing an intersection of solution spaces from a space containing both.

Step 1: List the two five-dimensional families. For $P$, the roots are $1$ twice, $-2$, and $\pm i$, giving $$y_P=(a+bx)e^x+ce^{-2x}+d\cos x+e\sin x.$$ For $Q$, the roots are $1$, $-2$ twice, and $\pm 2i$, giving $$y_Q=Ae^x+(B+Cx)e^{-2x}+E\cos 2x+F\sin 2x.$$ All constants in each display are independent.

Step 2: Retain only common modes. Exponential-polynomial modes at different exponents are independent. Equality of the two displays therefore removes both oscillatory pairs, $xe^x$, and $xe^{-2x}$. Precisely the simple shared modes survive: $$\boxed{y=Ae^x+Be^{-2x}.}$$ They are independent, so the common space has dimension two. In polynomial language, the common characteristic factor is the greatest common divisor $(r-1)(r+2)$: a mode must satisfy both multiplicity restrictions and hence the smaller one.

Step 3: Fit the common initial data. We need $A+B=3$ and $A-2B=0$, so $A=2,B=1$. Thus $$\boxed{y=2e^x+e^{-2x},\qquad y''(0)=6,\qquad y'''(0)=-6.}$$ Each exponential is a shared mode, proving both equations directly. The first two data fix the solution within the common space; higher data cannot be chosen independently there.

Step 4: Build the smallest operator containing both kernels. An operator annihilating every solution of either equation must include both repeated real-root chains and both oscillatory pairs. Its characteristic polynomial must be divisible by both $P$ and $Q$. Their monic least common multiple is $$\boxed{R(r)=(r-1)^2(r+2)^2(r^2+1)(r^2+4),\qquad \deg R=8.}$$ The multiplicity of each root is now the larger of its two multiplicities. This polynomial is sufficient by the characteristic-root rule and necessary because each listed mode must be killed. Thus its order is minimal and its monic choice unique. Its kernel contains both five-dimensional spaces, whereas the common space is their two-dimensional intersection.