Consider the following IVP.
{\displaystyle M(x,y)\,dx+N(x,y)\,dy=0}
is a homogeneous type if both functions
M (
x, y ) and
N (
x, y ) are
homogeneous functions of the same degree
n .
[1] That is, multiplying each variable by a parameter
{\displaystyle \lambda }, we find
{\displaystyle M(\lambda x,\lambda y)=\lambda ^{n}M(x,y)\,} and
{\displaystyle N(\lambda x,\lambda y)=\lambda ^{n}N(x,y)\,.}
Solution method
Introduce the change of variables
{\displaystyle y=ux}
;
{\displaystyle I(x,y)\,\mathrm {d} x+J(x,y)\,\mathrm {d} y=0,\,\!}
is called an
exact differential equation if there exists a
continuously differentiable function
F , called the
potential function , so that
{\displaystyle {\frac {\partial F}{\partial x}}=I}
and
{\displaystyle {\frac {\partial F}{\partial y}}=J.}
Solutions to exact differential equation
substitute g(y) into
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linear Equation and integrating factor
Proof:
{\displaystyle y’+P(x)y=Q(x)}
{\displaystyle M(x)}, integrating factor
{\displaystyle M(x)=e^{\int _{s_{0}}^{x}P(s)ds}}
then:
{\displaystyle {\begin{aligned}(1)\qquad &M(x){\underset {\text{partial derivative}}{(\underbrace {y’+P(x)y} )}}\\(2)\qquad &M(x)y’+M(x)P(x)y\\(3)\qquad &{\underset {\text{total derivative}}{\underbrace {M(x)y’+M'(x)y} }}\end{aligned}}}
Going from step 2 to step 3 requires that
{\displaystyle M(x)P(x)=M'(x)},
{\displaystyle {\begin{aligned}(4)\qquad &M(x)P(x)=M'(x)\\(5)\qquad &P(x)={\frac {M'(x)}{M(x)}}\\(6)\qquad &\int _{s_{0}}^{x}P(s)ds=\ln M(x)\\(7)\qquad &e^{\int _{s_{0}}^{x}P(s)ds}=M(x)\end{aligned}}}
Solution:
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Equations of Bernoulli,Ricatti, and Clairaut
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Linear dependence:
w=0 => Linearly Independent
linearly dependent => w=0
Reduction of order
Solution:
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HOMOGENEOUS LINEAR EQUATIONS
WITH CONSTANT COEFFICIENTS
Two distinct real roots
Two complex conjugate roots
One repeated real root
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Annihilator Operator