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dy = F(ax + by + c) into a separable dx 15. Show that the substitution v = ax + by + c transforms the differential equation equation. A first-order differential equation is called separable provided that f(x,y) can be written as (4) dv If v = ax + by + c, then = (2) dx dy Thus, for the substitution v = ax + by + c, = (3). dx dy dy = F(ax + by + c) to get (4) dx Substitute v and in the given differential equation dx Rearrange the equation found in the previous step to isolate the v and x terms. The rearranged equation is (5) = dx. This is a separable equation.

dy = F(ax + by + c) into a separable dx 15. Show that the substitution v = ax + by + c transforms the differential equation equation. A first-order differential equation is called separable provided that f(x,y) can be written as (4) dv If v = ax + by + c, then = (2) dx dy Thus, for the substitution v = ax + by + c, = (3). dx dy dy = F(ax + by + c) to get (4) dx Substitute v and in the given differential equation dx Rearrange the equation found in the previous step to isolate the v and x terms. The rearranged equation is (5) = dx. This is a separable equation.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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dy = F(ax + by + c) into a separable dx 15. Show that the substitutiony = ax + by + c transforms the differential equation equation. A first-order differential equation is called separable provided that f(x,y) can be written as (2) dv If v = ax + by + c, then = (2) dx dy Thus, for the substitution v = ax + by + c, = (3) dx dy dy = F(ax + by + c) to get (4) dx Substitute v and in the given differential equation dx Rearrange the equation found in the previous step to isolate the v and x terms. The rearranged equation is (5) = dx. This is a separable equation. (3) 1 ( dv dy + b. xp. (1) dy (2) + P(x)y = Q(x)y". dx a dx 1( dv - a b dx dy dy =f dx O a+b +C. dx dy + P(x)y = Q(x). dy O a+b- dx dv - a- dx dx dy O a-b. dx g(x) dv dy f(x,y)= g(x)k(y) = dx a- dx h(y) (4) dv - a-c = F(v). (5) dv b dx a- bF(v) dv dv = F(v). a- dx bF(v) + a dv 1 (dv - a| = F(v). aF(v) + b b dx dv 1 ( dv -b= F(v). a dx bF(v) + a + c
Transcribed Image Text:dy = F(ax + by + c) into a separable dx 15. Show that the substitutiony = ax + by + c transforms the differential equation equation. A first-order differential equation is called separable provided that f(x,y) can be written as (2) dv If v = ax + by + c, then = (2) dx dy Thus, for the substitution v = ax + by + c, = (3) dx dy dy = F(ax + by + c) to get (4) dx Substitute v and in the given differential equation dx Rearrange the equation found in the previous step to isolate the v and x terms. The rearranged equation is (5) = dx. This is a separable equation. (3) 1 ( dv dy + b. xp. (1) dy (2) + P(x)y = Q(x)y". dx a dx 1( dv - a b dx dy dy =f dx O a+b +C. dx dy + P(x)y = Q(x). dy O a+b- dx dv - a- dx dx dy O a-b. dx g(x) dv dy f(x,y)= g(x)k(y) = dx a- dx h(y) (4) dv - a-c = F(v). (5) dv b dx a- bF(v) dv dv = F(v). a- dx bF(v) + a dv 1 (dv - a| = F(v). aF(v) + b b dx dv 1 ( dv -b= F(v). a dx bF(v) + a + c
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