Example 1 : Solve : , given that is one integral. Solution : The given equation is Given equation (1) when compared with then It is given that is an integral belonging to CF then the complete solution is By putting in (1) it reduced to then …(3) So (3) reduced to On integration On …
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Example 1 : Apply the method of variation of parameter to solve Solution : We are given with …(1) Auxiliary equation So, …(2) So, So, two solutions and are independent Now, Hence Example 2 : Using the method of variation of parameter, solve Solution : We are given …(1) The roots of the auxiliary equation …
The method of variation of parameters can always be used to find a particular solution of the non-homogeneous linear differential equation …(1) We know the general solution …(2) Of associated homogeneous equation …(3) The basic idea of method of variation of parameters is that we replace, the constants, or parameter in the complementary function (2) …
Example 7 : Solve : Solution : Dividing by (x + 2), the given equation in standard form is …(1) Comparing (1) with we have …(1) Here, showing that a part of (1) is …(3) Let the general solution of (1) be y = uv …(4) Then u is given by or or …(5) Let …
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The given differential equation is of the forms …(1) where P, Q, R are as function of x alone Suppose y = u(x) is a solution of the equation …(2) The equation (2) is called the complementary function (CF) of equation (1) and u(x) is called an integral of CF y = u(x) is a …
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A linear differential equation of the form …(1) Where are constants and X is a function of x and is called Legendre’s linear equation. The solution of Legendre’s linear equation is same as the solution homogeneous linear equation, with a little difference. Here, we take And Now Where Then And So, Similarly, on substituting the …
Example 1 : Solve : Solution : The given equation …(1) is homogeneous linear equation We put Then and So, (1) reduced to …(2) It’s AE is Hence, the general solution will be Example 2 : Solve : Solution : The given equation …(1) is a homogeneous linear equation We put Then Then (1) reduced …
A linear differential equation of the form …(1) is called homogeneous linear differential equation. Where are constants, and X is either a constant or function of x only. The (1) can be solved as follows: We put …(2) Now where So, we can take …(3) Again or … (4) Proceeding in this way, we can …
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is a function of x Proof : Let …(1) Now consider …(2) …(3) …(4) So, Hence, Example 1 : Solve : Solution : The given equation is …(1) Its auxiliary equation is …(2) Now Hence, the complete solution is Example 2 : Solve : Solution : Given equation is Its auxiliary equation is …(1) Hence, …