= Solution
Assume independent samples $X_1,\ldots,X_n$ and $Y_1,\ldots,Y_m$, each consisting of independent <normal random variables>, with respective means $\mu_1,\mu_2$ and a common unknown <variance> $\sigma^2>0$. Take $n,m\geq2$. Normality, independence and equal variances are the assumptions giving the exact pooled <Student's t-test>; unequal variances would require a different test. We test $H_0:\mu_1=\mu_2$ against an unrestricted difference.
Put $N=n+m$ and
$$
W=\sum_i(X_i-\bar X)^2+\sum_j(Y_j-\bar Y)^2,\qquad
B=\frac{nm}{N}(\bar X-\bar Y)^2.
$$
The normal likelihood is proportional to $(\sigma^2)^{-N/2}\exp[-S/(2\sigma^2)]$, where $S$ is the residual sum of squares. Under the unrestricted model, the maximizing means are $\bar X,\bar Y$ and $S=W$. Under $H_0$, the maximizing common mean is $(n\bar X+m\bar Y)/N$ and decomposition about the <sample means> gives $S=W+B$. For either model, maximizing over <variance> gives $\widehat\sigma^2=S/N$. Thus the <generalized likelihood-ratio test> statistic is
$$
\Lambda=\left(\frac{W}{W+B}\right)^{N/2}
=\left(1+\frac{T^2}{N-2}\right)^{-N/2},\qquad
T=\frac{\bar X-\bar Y}{s_p\sqrt{1/n+1/m}},\quad s_p^2=\frac{W}{N-2}.
$$
Small $\Lambda$ is therefore equivalent to large $|T|$.
Under the null, $Z=(\bar X-\bar Y)/[\sigma\sqrt{1/n+1/m}]$ is standard normal. In each sample an orthogonal change of Gaussian coordinates separates the sample-mean coordinate from the centered residual coordinates; those coordinates are independent. Consequently $V=W/\sigma^2$ has a <chi-squared distribution> with $N-2$ degrees of freedom and is independent of $Z$. Therefore $T=Z/\sqrt{V/(N-2)}$ has the <Student's t-distribution> with $N-2$ degrees of freedom. \b[At significance level $\alpha$, reject exactly when $\boxed{|T|>t_{N-2,1-\alpha/2}}$], the two-sample pooled t-test.
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