Problem: Use the substitution to solve the differential equation .
Solution: Well, let’s do what it says. From the chain rule, we have
Then we also have by the product rule and chain rule again
So we can make the substitutions
to obtain the differential equation
But we know the solution to this is so our final solution is
Comment: This substitution is, of course, not really natural but was actually found after solving the ODE in another way. Fortunately, it simplifies the problem rather greatly and seems to be a useful technique to look out for.
Practice Problem: Find another way to solve the differential equation.
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