Substitutions

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Equations of the form: dydx=G(ax+by+c)

Method for Solving Equations in the Form dydx=G(ax+by+c)

  1. Use the substitution z=ax+by+c.

  2. Differentiate using product rule: dzdx=a+bdydx This is the second substitution: dydx=dzdxab.

  3. Rewrite the DE as dzdxab=G(z) or dzdx=bG(z)+a.

  4. The DE is now separable: 1bG(z)+adz=dx.

  5. Rewrite the solution in terms of ax+by+c.

  6. Identify missing solutions or solutions that are not solutions of the original DE.

Question 02

Solve the following DEs:

  1. dydx=(x+y4)2

  2. dydx=(16x+y+3)4

  3. dydx=tan2(x+y)

  4. dydx=(xy5)2, y(1)=1

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Last Modified: Sunday, 6 September 2020 14:15 EDT