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| Solution of a Differential Equation |
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| The functional relation-ship between the independent variable and the dependent variable (such as y = f(x)) which satisfies the given differential equaion is called the solution of the differential equation. |
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| Example: |
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| Consider the function |
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| f(x) = x2 + Ax + B |
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Clearly  (x) is a solution of the differential equation. |
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| y''= 2, because if the second derivative of
f(x) gives 2. That is |
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| y = x2 + Ax + B is a general solution of the differential equation. |
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| y'' = 2 |
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Each value of A and each value of B (A, B  R) gives a particular solution of the differential equation y'' = 2. |
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| Particular solution of a differential equation |
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| A solution obtained, by assigning particular values to the arbitrary constants in the general solution of the differential equation, is called its particular solution. |
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| General solution of a differential equation |
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| If the solution of a differential equation of order n contains n arbitrary constants, then it is called the General solution of the differential equation. |
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| Suppose the function |
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| Then |
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| \ f(x) = x2 + 2x + 1 is a particular solution of the differential equation y''= 2. |
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| For the function |
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| Because of these condition, the 2nd order differential equation y''= 2 has particular solution x2 + x + 2. |
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| The values f(0) = 2 and f'(0) = 1 are called initial values. |
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| The problem of finding the solution of a differential equation that satisfies these prescribed initial conditions is known as an Initial value problem. |
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| Example: |
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Show that the function  = e-x+ 2 is the solution of the differential equation y'+ y = 2 , y(0) = 3. |
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| Solution: |
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If we replace y by  , we have |
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| y'+ y |
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| \ f(x) satisfies the equation y' + y = 2 |
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Moreover,  |
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