Linear Independence & the Wronskian
Differential Equations · Section 3.2 · generated practice set
Wronskians, fundamental sets, Abel's theorem, and finding the largest interval on which a solution is guaranteed.
Practice this set → Fresh numbers on every attempt. No account needed.
Method
Three things get asked, and they are connected.
1. Compute a Wronskian
$$W(y_1,y_2)(t) = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2'\end{vmatrix} = y_1y_2' - y_1'y_2$$
Worth memorizing: $W(e^{r_1t}, e^{r_2t}) = (r_2 - r_1)e^{(r_1+r_2)t}$, $W(t^m,t^n) = (n-m)t^{m+n-1}$, $W(\cos\omega t, \sin\omega t) = \omega$.
2. Fundamental sets
If $y_1,y_2$ both solve the same second-order linear homogeneous equation and $W \neq 0$ at some point, they form a fundamental set and every solution is $c_1y_1 + c_2y_2$.
3. Interval of existence
Put $P(t)y'' + Q(t)y' + R(t)y = G(t)$ into standard form by dividing by $P$. The solution of the IVP is guaranteed on the largest interval containing $t_0$ where $Q/P$, $R/P$, $G/P$ are all continuous — that is, where $P \neq 0$ and nothing else breaks. The initial values never affect the interval; only $t_0$ does.
Abel's theorem
$W(t) = c\,e^{-\int p(t)\,dt}$, so $W$ is either identically zero or never zero on the interval.
Definitions and theorems
Worked example
Find the longest interval on which $(t^2 - 9)y'' + 2ty' + \ln(t)\,y = 0$, $y(4) = 1$, $y'(4) = 0$ is certain to have a unique solution.
- Standard form: divide by $t^2 - 9$, which vanishes at $t = \pm 3$.
- $p(t) = \frac{2t}{t^2-9}$ and $q(t) = \frac{\ln t}{t^2-9}$.
- $\ln t$ additionally requires $t > 0$.
- Break points: $t = 0$ (from $\ln t$) and $t = 3$ (the relevant zero of $t^2 - 9$); $t = -3$ is already excluded by $t>0$.
- $t_0 = 4$ lies in $(3, \infty)$, and everything is continuous there. Answer: $(3, \infty)$.
Common mistakes
- Forgetting to divide by the leading coefficient. The singular points come from $P(t) = 0$, which you only see after dividing.
- Letting the initial values change the interval. Only $t_0$ matters for which piece you take.
- Including the break point. Intervals are open: $(3,\infty)$, never $[3,\infty)$.
- Wronskian sign/order. $W = y_1y_2' - y_1'y_2$ — swapping $y_1$ and $y_2$ flips the sign, which is harmless for independence but not for a numeric answer.
- Concluding dependence from $W(t_0) = 0$ for arbitrary functions. Abel's dichotomy applies to solutions of the same linear ODE, not to any two functions.
Practice it
Reading the method is not the same as being able to run it under time pressure. This set generates a new problem with new coefficients every attempt, marks each part separately, and shows the full worked solution.