Solving First-Order Differential Equations

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Solving First-Order Differential Equations

• A first-order Diff. Eq. In x and y is separable
  if it can be written so that all the y-terms are
  on one side and all the x-terms are on the other

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First-Order Differential Equations

• A differential equation has variables separable
  if it is in one of the following forms
• Integrating both sides, the general solution
  will be

dy f(x) dx g(y)
OR
g(y)dy - f(x)dx 0

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Separable Differential Equations
Another type separable differential equation can
be expressed as the product of a function of x
and a function of y.
Example 1
Multiply both sides by dx and divide both sides
by y2 to separate the variables. (Assume y2
is never zero.)
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Separable Differential Equations
Another type of separable differential equation
can be expressed as the product of a function of
x and a function of y.
Example 1
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Example 2
Separable differential equation
Combined constants of integration
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Example 2
We now have y as an implicit function of x.
We can find y as an explicit function of x by
taking the tangent of both sides.
Notice that we can not factor out the constant C,
because the distributive property does not work
with tangent.
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Example 3Differential equation with initial
condition These are called Initial value
problems

• Solve the differential equation dy/dx -x/y
  given the initial condition y(0) 2.
• Rewrite the equation as ydy -xdx
• Integrate both sides solve
• Since y(0) 2, we get 4 0 C, and therefore
• x2 y2 4

y2 x2 C where C 2k
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Example 4 Solve
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Solution to Example 4