When a mathematician solves a differential equation, they are finding {{CMD_364}}functions{{CMD_365}} satisfying the equation. {{CMD_366}} Which of the following functions solve the differential equation {{MATH_367}} {{CMD_368}} {{CMD_369}}{{MATH_370}}{{CMD_371}} {{CMD_372}}{{MATH_373}}{{CMD_374}} {{CMD_375}}{{MATH_376}}{{CMD_377}} {{CMD_378}}{{MATH_379}}{{CMD_380}} {{CMD_381}}{{MATH_382}}{{CMD_383}} {{CMD_384}}{{MATH_385}}{{CMD_386}} {{CMD_387}} {{CMD_388}} {{CMD_389}} We can directly check that any function {{MATH_390}} is a solution to our differential equation {{MATH_391}}. Could there be any others? It turns out that these are the {{CMD_392}}only{{CMD_393}} solutions. But showing that we didn't miss any is a bit tricky. {{CMD_394}} {{CMD_395}} Well, suppose we have some mysterious function {{MATH_396}} and all we know is that {{MATH_397}}. Let's define a new function {{MATH_398}}. Since our denominator is never 0, the quotient rule tells us that {{MATH_399}} But we know that a function with a 0 derivative is constant, so {{MATH_400}}. Plugging this back into our formula for {{MATH_401}} tells us that {{MATH_402}}.