Definition 2.1 separable equation
A first-order differential equation of the form
dy g ( x ) h( y ) dx
is said to be separable or to have separable variables.
Information about the nature of solutions directly from The differential equation itself. Integral curves
dy 1 一阶微分方程 f ( x, y )的解y ( x)代表xy平面上的 dx 一条曲线，称为微分方程的积分曲线。
方向场。
在方向场中，方向相同的点的几何轨迹为等斜线。
dy f ( x, y )的等斜线方程为f ( x, y ) （参数）,给出k k dx
充分接近的值，就可得到足够密集的等斜线族。
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2.2 separable variables
Solution by integration
x 0, y 1 into the ordinary solution in order to define
arbitrary constant, that
c 1
Therefore, the particular solution is
1 y 1 sin x
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(1)
also can transform to the separate variable equation, a1 , a2 , b1 , b2 , c1 , c2 separately are constants. We discuss it by three cases.
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When f is independent of the variable y -that is, f ( x, y) g ( x) The differential equation
dy g ( x) (1) dx can be solved by integration. If g(x) is a continuous
y du dy u y ux x u x dx dx
du x u g (u ) du g (u ) u dx x dx
This is a separable equation.
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if
g (u) u 0
The above equation is a separable equation. If exist u=u0, , and g(u0)- u0 =0. We can have the result u=u0, is a solution of the above equation , so y= u0 is a solution of the previous equation.
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Example 1 solve the equation dy y y tg dx x x Solution Let
y dy du u x u x dx dx
put the above into the equation, get du x u u tgu dx du tgu dx x Let the above separate variable and integrate,
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dy dx g ( x)h( y)
Observe that by dividing by the function h(y) we can write a separable equation as
1 dy g ( x) dx h( y )
and therefore
1 h( y) dy g ( x)dx
Example 3
find the general solution of dy P ( x) y (2) dx P(x) is continuous function of x. Solution separate the variables, get
dy p( x)dx y
integrated by the both sides, There,
y sin u cx sin cx x
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Exercise solve the equation
dy x 2 xy y ( x 0) dx
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2) the form
dy a1 x b1 y c1 dx a2 x b2 y c2
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Example 2
solve the initial-value problem
dy y 2 cos x, y (0) 1 dx
Solution please separate the variables
dy cos xdx 2 y
Integrated by the both, get 1 sin x c y from which it follows that, the general solution
a1 b1 0 and c1 , c2 are not all zero. a2 b2
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而通解y ( x, c)对应于xy平面上的一族曲线(积分曲线
族)满足y( x0 ) y0条件的特解是过( x0 , y0 )的一条积分曲线。
2.设f ( x, y)的定义域为D，在( x, y) D处画上一个小线段， 其斜率等于f ( x, y)，带这种直线段的区域D为方程的
2 first-order differential equation
2.1 Solution curves without the solution
2.2 separable variables 2.3 linear equations
2.4 exact equation
2.5 solutions by substitutions
where H(y) and G(x) are antiderivatives of 1 h( y) and g(x), respectively.
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Methoer family of solutions, usually given
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2.1 Solution curves without the solution If we can neither find nor invent a method for solving it
analytically, it is often possible to glean useful
~ c
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~ ln y p( x)dx c
is an arbitrary constant.
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From the define of log, we get
p ( x ) dx c y ec e p ( x ) dx y e
Let e c , have
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Have
dx ctgudu x
ln sin u ln x c
sin u e x
c
sin u cx (e c)
c
Otherwise, the equation have solution tgu=0 , that is to say sinu=0 , therefore the solution is
function, then integrating both sides of (1) gives the
solution
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y g ( x)dx G ( x) c
where G(x) is an antiderivative (indefinite integral) of g(x).
variable equation.
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(2) the form
a1 a2
a1 b1 b1 0 a2 b2 b2
a1 b1 k a2 b2
Suppose the ratio is k, that is to say
Then the equation can be
1 y sin x c
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c is an arbitrary constant
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Otherwise, the equation also have the solution
y 0
In order to determine the particular solution, put
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2
reduction to separation of variables
dy y g( ) dx x
We only introduce two types 1) the form
Is called homogeneous equation, g(u) is continuous function of u.