Question

[Tutorial] differential equation

Answer 1

Factors of Integration

Not all Differential equations is an exact. The easiest solution when Differential equation is an exact is BREAK TROUGHT.


for example:
\[(3y + 4xy^{2}) dx + ( 2x + 3x^{2} y) dy = 0\]
\[M = 3y + 4xy^{2} \rightarrow \frac{ dM }{ dy } = 3 + 8xy\]
\[N = 2x + 3x^{2} \rightarrow \frac{ dN }{ dx } = 2 + 6xy\]
where, when \(\frac{ dM }{ dy } \neq \frac{ dN }{ dx } \), differential equation is no exact.
(cannot be solved by the method of integral and grouping)

SOP:
a) make exact differential equation:
\[\frac{ dy }{ dx } + P(x) y = Q (x)\]
b) An integral factor \(\mu (x,y)\), the initial differential equation is multiplied by an integral factor.
\[\mu \frac{ dy }{ dx } + \mu P(x)y = \mu Q(x)\]
\(\mu = e^{\int\limits P(x)dx}\) substitutes into differential equations

for example:
\[(3y + 4xy^{2}) dx + (2x + 3x^{2}y) dy = 0\]
\[dy + \frac{ 3y + 4xy^{2} }{ 3x^{2}y } dx = 0\]

\[\frac{ dy }{ dx } + \frac{ 3y + 4xy^{2} }{ 2x + 3x^{2}y } = 0\]

Answer 2

use separation of variable technique

General form:
\[F (x) G(y) dx + f(x) g(y) dy = 0\]
with an integration factor
\[\mu = \frac{ 1 }{ f(x) G(y)} \]
\[\frac{ F(x) }{ f(x) } dx + \frac{ g(y) }{ G(y)dy } = 0\]
excat because:
\[\frac{ d }{ dy } (\frac{ F(x) }{ f(x) }) = \frac{ d }{ dx } (\frac{ g(y) }{ G(y) })\]