# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

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Serious Substitution. Topic: Calculus. October 25th, 2006

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

and

to obtain the differential equation

.

But we know the solution to this is so our final solution is

.

QED.

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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.

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Practice Problem: Find another way to solve the differential equation.

Posted in Calculus || 3 Comments »

### 3 Responses to “Serious Substitution. Topic: Calculus.”

1. t0rajir0u Says:

Series. Lol.

2. Xuan Says:

another way.. well there’s the limit defination, and also the fundemental therome of calculus.
but there’s the d^2y/dx^2, so i don’t know if it works.

are any of these questions likely to appear on the SAT? or.. are there even any calculus on the SAT?