Substitutions and Transformations

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

Expected Educational Results

Homogeneous Equations

Definition: Homogeneous Equations

If the right-hand side of the equation dydx=f(x,y) can be written as a function of the ratio yx alone, i.e., dydx=f(yx), then the equation is homogeneous.

Method for Solving Homogeneous Equations}

  1. Identify if the DE is homogeneous, i.e., if and only if f(tx,ty)=f(x,y).

  2. Rewrite the DE as dydx=f(yx).

  3. Use the substitution ν=yx:

    • Rewrite as y=νx;

    • Differentiate using product rule: dydx=ν+xdνdx which is the second substitution.

  4. Rewrite the DE as ν+xdνdx=G(ν).

  5. The DE is now separable: 1G(ν)νdν=1xdx.

  6. Rewrite the solution in terms of x and y.

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

Question 01

Solve the following DEs:

  1. dydx=x2+y2xy

  2. dydx=y(xy)x2

  3. dydx=xyx+y

  4. (xy)dx+xdy=0

  5. (x2y2)dx+2xydy=0, y(1)=1

  6. dydx=x2xy+y2xy, y(1)=0

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