Let be a positive function such that the area bounded by from to is . Then the differential equation, whose general solution is , where and are arbitrary constants, is

Question

Let  be a positive function such that the area bounded by  from  to  is . Then the differential equation, whose general solution is , where  and  are arbitrary constants, is

Canvas Classes · Accepted Answer

🧠 Differentiate the Area to Find , Then Form the ODE

Read: . Differentiate by Leibniz: . Renaming: .

General solution:  — two arbitrary constants. So the ODE is 2nd order.

Differentiate: , . Eliminate : , i.e. .

Compute , .

🗺️ Working It Out

... wait,  means , i.e. .

Multiply by : , or .

Answer: (d).

⚡ The Smart Move — Eliminating  Yields the ODE

For  with two constants, the ODE is 2nd order with the form  (the Wronskian-style relation eliminating both constants).

⚠️ Where Students

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