Equations with Linear Coefficients

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

Expected Educational Results

Equations with Linear Coefficients

NOTE: You will not be assessed on ODEs with linear coefficients.

Definition: Equations with Linear Coefficients

Equations with linear coefficients is an ODE in the form:

(a1x+b1y+c1)dx+(a2x+b2y+c2)dy=0

Method for Solving Equations with Linear Coefficients

Case 1: a1b2=a2b1
  1. Substitute v=a1x+b1ya1 which implies a1(vx)=b1y and a2b1b2(vx)=b1y since a1b2=a2b1. a2b1b2(vx)=b1y further simplifies to a2(vx)=b2y.

  2. Solve a2(vx)=b2y for v: v=a2x+b2ya2.

  3. Differentiate the substitution expression: dvdx=1+b1a1dydx.

  4. Rewrite the derivative above into: dydx=a1b1(dvdx1).

  5. Substitute the formulas for v from steps 1 and 2 and dydx from step 3 into dydx=a1x+b1y+c1a2x+b2y+c2 to obtain: a1b1(dvdx1)=a1v+c1a2v+c2

  6. Solve for dvdx: dvdx=1+b1a1(a1v+c1a2v+c2).

  7. The DE is now separable into: dv1+b1a1(a1v+c1a2v+c2)=dx.

Case 2: a1b2a2b1 and c1=c2=0
  1. Rewrite (a1x+b1y)dx+(a2x+b2y)dy=0 into dydx=a1x+b1ya2x+b2y=a1+b1yxa2+b2yx.

  2. The DE is now homogeneous.

Case 3: a1b2a2b1 and both c10 and c20
  1. Substitute x=u+h and y=v+k, where h and k are constants.

  2. Solve the system: a1h+b1k+c1=0, a2h+b2k+c2=0 for h and k.

  3. Rewrite the DE as dvdu=a1u+b1va2u+b2v=a1+b1vua2+b2vu.

  4. The DE is now homogeneous.

Question 04

Solve the following DEs:

  1. dydx=x2y22x+y+6

  2. dydx=2x+yyx

  3. dydx=x+3y+44y3x+1

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