Ch 04/Equations Reducible to Homogeneous Form/H.K. Dass 3.8

Equations Reducible to Homogeneous Form

Equations like dy/dx = (ax+by+c)/(Ax+By+C) look almost homogeneous, but constants c and C spoil things. A coordinate shift or substitution fixes this.

ITwo Cases

Case I: $a/A \neq b/B$

Substitute , . Choose so that and . This removes the constants, giving a homogeneous equation.

Case II: $a/A = b/B$ (the "failure" case)

The above fails because become infinite. Put and use separation of variables.

Note

Case II occurs when the two lines are parallel. Since parallel lines never intersect, we cannot find a finite .

Example 1 (Case I)medium

Solve .

Show step-by-step solution+

Check

. Case I.

Find h, k

Solving: . Put .

Homogeneous equation

. Put , use partial fractions.

Final answer

Example 2 (Case II)medium

Solve .

Show step-by-step solution+

Rewrite

. Here . Case II.

Put z = x + 2y

Separate and integrate

Final answer

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