Definition of homogeneous differential equation with example
Answers
A first‐order differential equation is said to be homogeneous if M(x,y) and N(x,y) are both homogeneous functions of the same degree. Example 6: The differential equation. is homogeneous because both M(x,y) = x 2 – y 2 and N(x,y) = xy are homogeneous functions of the same degree (namely, 2).
Homogeneous differential equations are those where f(x,y) has the same solution as f(nx, ny), where n is any number. They typically cannot be solved as written, and require the use of a substitution. The general form of a homogeneous differential equation is . To solve the equation, use the substitution . Using the substitution and the product rule, the equation becomes . The equation is rearranged, simplified, and separated:
; then ; then ; then
(If the resulting equation cannot be separated, the original equation was not homogeneous, or an error was made while solving the equation.) Once the separated equation is solved, replace the original equation using the previous substitution.
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