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Magazine Chapter 4: Problem 19
\(y^{\prime \prime}+x y^{\prime}=\sin x, y(0)=1, y^{\prime}(0)=0\)
Short Answer
Expert verified
Identify the equation type, solve for the homogeneous equation, find a particular solution, combine solutions, then apply initial conditions to find the specific solution.
Step by step solution
01
Identify the type of differential equation
Recognize that this is a second-order linear non-homogeneous differential equation of the form: \[y'' + p(x)y' + q(x)y = g(x)\]Here, \[p(x) = x, q(x) = 0, g(x) = \sin x\]
02
Find the complementary solution
Solve the homogeneous equation \[y'' + x y' = 0\].Assume a solution of the form \[y_h = C_1 + C_2e^{\frac{-x^2}{2}}\].To find the complementary solution, solve the associated homogeneous problem.
03
Find a particular solution
To solve the non-homogeneous part, \[y_p'' + x y_p' = \sin x\], use the method of undetermined coefficients or variation of parameters to propose a solution that incorporates \( \sin x \).
04
Combine the solutions
Combine the complementary and particular solutions to find the general solution:\[y = y_h + y_p = C_1 + C_2e^{\frac{-x^2}{2}} + y_p\].
05
Apply initial conditions
Use the initial conditions \(y(0)=1\) and \(y'(0)=0\) to determine the constants \(C_1\) and \(C_2\).Substitute these values back into the general solution.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
complementary solution
The complementary solution, often denoted as \( y_h \), is the solution to the homogeneous differential equation. For our example, we start with the equation:
\[ y'' + xy' = 0 \]
Assuming a solution of the form \( y_h = C_1 + C_2e^{\frac{-x^2}{2}} \), we solve for the homogeneous problem. The complementary solution represents the general behavior of the differential equation without considering the non-homogeneous term like \( \sin x \) in the original equation. By solving the above equation, we obtain:
\[ y_h = C_1 + C_2e^{\frac{-x^2}{2}} \]
where \( C_1 \) and \( C_2 \) are constants that will be determined through initial conditions.
\[ y'' + xy' = 0 \]
Assuming a solution of the form \( y_h = C_1 + C_2e^{\frac{-x^2}{2}} \), we solve for the homogeneous problem. The complementary solution represents the general behavior of the differential equation without considering the non-homogeneous term like \( \sin x \) in the original equation. By solving the above equation, we obtain:
\[ y_h = C_1 + C_2e^{\frac{-x^2}{2}} \]
where \( C_1 \) and \( C_2 \) are constants that will be determined through initial conditions.
particular solution
To solve the non-homogeneous differential equation, we need to find a particular solution \( y_p \). The particular solution addresses the extra term, \( \sin x \), in the original equation:
\[ y'' + xy' = \sin x \]
There are different methods to find \( y_p \), with 'method of undetermined coefficients' and 'variation of parameters' being the most commonly used. Using one of these methods, we propose a specific form for \( y_p \) that incorporates \( \sin x \). For instance, if we suppose a form that involves the sine function directly, we might propose a solution and test it to ensure it satisfies the original differential equation.
\[ y'' + xy' = \sin x \]
There are different methods to find \( y_p \), with 'method of undetermined coefficients' and 'variation of parameters' being the most commonly used. Using one of these methods, we propose a specific form for \( y_p \) that incorporates \( \sin x \). For instance, if we suppose a form that involves the sine function directly, we might propose a solution and test it to ensure it satisfies the original differential equation.
initial conditions
Initial conditions are critical in finding the unique solution to a differential equation. In our example:
\[ y(0) = 1, y'(0) = 0 \]
These conditions mean that when \( x = 0 \), the function \( y \) is 1, and its first derivative is 0. By applying these initial conditions to our combined solution, which includes both the complementary and particular solutions, we can solve for the constants \( C_1 \) and \( C_2 \).Substituting these values back into the general solution provides a specific solution that satisfies the initial conditions.
\[ y(0) = 1, y'(0) = 0 \]
These conditions mean that when \( x = 0 \), the function \( y \) is 1, and its first derivative is 0. By applying these initial conditions to our combined solution, which includes both the complementary and particular solutions, we can solve for the constants \( C_1 \) and \( C_2 \).Substituting these values back into the general solution provides a specific solution that satisfies the initial conditions.
method of undetermined coefficients
The 'method of undetermined coefficients' is a technique to find the particular solution to a non-homogeneous differential equation. This method involves proposing a form for \( y_p \) based on the non-homogeneous term \( g(x) \). In our example, \( g(x) = \sin x\):
1. **Guess a form for \( y_p \)**: Since \( g(x) \) is \( \sin x \), we guess a function that might include sine or cosine, depending on the complexity.
2. **Determine coefficients**: Through differentiation, plug the guessed \( y_p \) back into the equation to figure out the unknown coefficients. Methodically solve for these coefficients so that \( y_p \) satisfies the original non-homogeneous equation.
1. **Guess a form for \( y_p \)**: Since \( g(x) \) is \( \sin x \), we guess a function that might include sine or cosine, depending on the complexity.
2. **Determine coefficients**: Through differentiation, plug the guessed \( y_p \) back into the equation to figure out the unknown coefficients. Methodically solve for these coefficients so that \( y_p \) satisfies the original non-homogeneous equation.
variation of parameters
Another approach to finding a particular solution is 'variation of parameters'. This is especially useful for more complex non-homogeneous equations. For our equation:
\[ y'' + xy' = \sin x \]
we follow these steps:
1. **Solve the homogeneous equation**: First, identify the complementary solution \( y_h \).
2. **Formulate the particular solution**: Propose a particular solution where the parameters are functions rather than constants. This lets us vary them to fit the non-homogeneous term.
3. **Substitute and solve**: Plug the proposed solution into the original equation. Solve the resulting system of equations to find the functions. This way, we find a particular solution that together with the complementary solution forms the general solution.
\[ y'' + xy' = \sin x \]
we follow these steps:
1. **Solve the homogeneous equation**: First, identify the complementary solution \( y_h \).
2. **Formulate the particular solution**: Propose a particular solution where the parameters are functions rather than constants. This lets us vary them to fit the non-homogeneous term.
3. **Substitute and solve**: Plug the proposed solution into the original equation. Solve the resulting system of equations to find the functions. This way, we find a particular solution that together with the complementary solution forms the general solution.
