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6-2 Solving Differential Equations

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Presentation on theme: "6-2 Solving Differential Equations"β€” Presentation transcript:

1 6-2 Solving Differential Equations
Rizzi – Calc BC

2 Topics Solving by separation of variables Proportionality
Finding Particular Solutions

3 Separation of Variables
Steps to solving a differential equation: Separate variables Integrate both sides That’s it. X only Y only Both 𝑑𝑦 𝑑π‘₯ =5βˆ’8π‘₯ 𝑑𝑦 𝑑π‘₯ =6βˆ’π‘¦ 𝑦′=π‘₯(1+𝑦)

4 Challenge Try this one: π‘₯𝑦+ 𝑦 β€² =100π‘₯

5 Proportionality If two things are proportional, they are related by a constant of proportionality Example: Hooke’s Law says 𝐹=βˆ’π‘˜π‘₯ Force is proportional to displacement, (𝐹~π‘₯) when considering a constant of proportionality that depends on the material of the spring

6 Direct vs. Inverse Proportionality
Directly Proportional Inversely Proportional 𝑦~π‘₯ Or 𝑦=π‘˜π‘₯ Ex. The rate of change of P with respect to t is inversely proportional to the square root of t 𝑦~ 1 π‘₯ Or 𝑦= π‘˜ π‘₯ Ex. The rate of change of M with respect to x is proportional to 25-t

7 Finding Particular Solutions
Just like what we did before. Just more practice! Find the particular solution to the differential equation 𝑦 β€² = 2π‘₯ 𝑦 that goes through the point (2, 5)

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