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Differential Equations and Separation 1

الكلية كلية الهندسة     القسم  الهندسة الميكانيكية     المرحلة 1
أستاذ المادة احمد كاظم حسين الحميري       13/07/2018 14:37:17
Differential equations
Navier–Stokes differential equations used to simulate airflow around an obstruction
Navier–Stokes differential equations used to simulate airflow around an obstruction.
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In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.

Contents
1 Ordinary differential equations (ODE)
1.1 Alternative notation
1.2 Example
2 Partial differential equations
2.1 Example: homogeneous case
2.2 Example: nonhomogeneous case
2.3 Example: mixed derivatives
2.4 Curvilinear coordinates
3 Matrices
4 References
5 External links
Ordinary differential equations (ODE)
Suppose a differential equation can be written in the form

{\displaystyle {\frac {d}{dx}}f(x)=g(x)h(f(x))} {\frac {d}{dx}}f(x)=g(x)h(f(x))
which we can write more simply by letting {\displaystyle y=f(x)} y=f(x):

{\displaystyle {\frac {dy}{dx}}=g(x)h(y).} {\frac {dy}{dx}}=g(x)h(y).
As long as h(y) ? 0, we can rearrange terms to obtain:

{\displaystyle {dy \over h(y)}=g(x)\,dx,} {\displaystyle {dy \over h(y)}=g(x)\,dx,}
so that the two variables x and y have been separated. dx (and dy) can be viewed, at a simple level, as just a convenient notation, which provides a handy mnemonic aid for assisting with manipulations. A formal definition of dx as a differential (infinitesimal) is somewhat advanced.

Alternative notation
Those who dislike Leibniz s notation may prefer to write this as

{\displaystyle {\frac {1}{h(y)}}{\frac {dy}{dx}}=g(x),} {\frac {1}{h(y)}}{\frac {dy}{dx}}=g(x),
but that fails to make it quite as obvious why this is called "separation of variables". Integrating both sides of the equation with respect to {\displaystyle x} x, we have

{\displaystyle \int {\frac {1}{h(y)}}{\frac {dy}{dx}}\,dx=\int g(x)\,dx,\qquad \qquad (1)} \int {\frac {1}{h(y)}}{\frac {dy}{dx}}\,dx=\int g(x)\,dx,\qquad \qquad (1)
or equivalently,

{\displaystyle \int {\frac {1}{h(y)}}\,dy=\int g(x)\,dx} \int {\frac {1}{h(y)}}\,dy=\int g(x)\,dx
because of the substitution rule for integrals.

If one can evaluate the two integrals, one can find a solution to the differential equation. Observe that this process effectively allows us to treat the derivative {\displaystyle {\frac {dy}{dx}}} {\frac {dy}{dx}} as a fraction which can be separated. This allows us to solve separable differential equations more conveniently, as demonstrated in the example below.

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