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Elementary Differential Equations : Substitution Methods and Exact Equations

Substitution Methods and Exact Equations

Homogeneous Equations:
Dy/dx = F(y/x)

v = y/x, y = vx, dy/dx = v + x(dv/dx)

x(dv/dx) = F(v) - v

Bernoulli Equations:

dv/dx + (1-n) P(x)v = (1-n) Q(x)

Criterion for Exactness:

F(x,y) = ∫ M(x, y) dx + g(y)

Verify that the given differential equation is exact; then solve it.

31. (2x + 3y) dx + (3x + 2y) dy = 0

Answer: x2 + 3xy +y2 = C

35. (x3 + y/x) dx + (y2 + lnx) dy = 0

Answer: 3x4 +4y3 +12ylnx = C

38. (x + tan-1y) dx + ((x +y) / (1 + y2)) dy = 0

Answer: x2 + 2xtan-1y + ln(1 +y2) = C

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Substitution Methods and Exact Equations

Homogeneous Equations:
Dy/dx = F(y/x)

v = y/x, y = vx, dy/dx = v + x(dv/dx)

x(dv/dx) = F(v) - v

Bernoulli Equations:

dv/dx + (1-n) P(x)v = (1-n) Q(x)

Criterion for Exactness:

F(x,y) = ∫ M(x, y) dx + g(y)

Verify that the given differential equation is exact; then solve it.

31. (2x + 3y) dx + (3x + 2y) dy = 0

Answer: x2 + 3xy +y2 = C

Solution. This differential equation is in the form of P(x,y)dx+Q(x,y)dy=0, where P(x,y)=2x+3y and Q(x,y)=3x+2y.

Note that . So, there exists a function Z(x,y) such that dZ=P(x,y)dx+Q(x,y)dy. Hence, the general solution to P(x,y)dx+Q(x,y)dy=0 is

Z(x,y)=C, where C is a constant.

Now the question becomes to find such function Z(x,y). We can do it as follows.

As , we can integrate it with respect to x to ...

Solution Summary

Differential equatiosn are solved. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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