1. Consider the equation -xy"+y' + (x-1)y=0, x > 0. We can easily verify that y₁(x) = e* is a solution of the equation. Use reduction of order to determine the general solution of the equation.
1. Consider the equation -xy"+y' + (x-1)y=0, x > 0. We can easily verify that y₁(x) = e* is a solution of the equation. Use reduction of order to determine the general solution of the equation.
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![### Differential Equations: Reduction of Order
#### Example Problem:
1. Consider the equation \( -xy'' + y' + (x - 1)y = 0 \), where \( x > 0 \). We can easily verify that \( y_1(x) = e^x \) is a solution of the equation. Use reduction of order to determine the general solution of the equation.
To solve this differential equation, we will use the method of reduction of order. This technique is useful when one solution to a second-order linear differential equation is already known. In this case, we know that \( y_1(x) = e^x \) is a solution.
Let's assume a second solution of the form
\[ y_2(x) = v(x)e^x \]
where \( v(x) \) is a function to be determined. By substituting \( y_2(x) \) into the original differential equation and simplifying, we can find \( v(x) \) and thus the general solution.
### Steps to Solve Using Reduction of Order:
1. **Substitute** \( y_2(x) = v(x)e^x \) into the differential equation.
2. **Differentiate** to find expressions for \( y'_2 \) and \( y''_2 \).
3. **Simplify** the resulting equation, using the fact that \( e^x \) is a known solution.
4. **Solve** for \( v(x) \).
5. **Integrate** to find the general solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24e9c030-c4e6-49da-b6c6-4b7d6c7f6fcd%2F4424f660-1e6b-45ba-867f-3b1e5b52a0f6%2Fiai9rrp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Differential Equations: Reduction of Order
#### Example Problem:
1. Consider the equation \( -xy'' + y' + (x - 1)y = 0 \), where \( x > 0 \). We can easily verify that \( y_1(x) = e^x \) is a solution of the equation. Use reduction of order to determine the general solution of the equation.
To solve this differential equation, we will use the method of reduction of order. This technique is useful when one solution to a second-order linear differential equation is already known. In this case, we know that \( y_1(x) = e^x \) is a solution.
Let's assume a second solution of the form
\[ y_2(x) = v(x)e^x \]
where \( v(x) \) is a function to be determined. By substituting \( y_2(x) \) into the original differential equation and simplifying, we can find \( v(x) \) and thus the general solution.
### Steps to Solve Using Reduction of Order:
1. **Substitute** \( y_2(x) = v(x)e^x \) into the differential equation.
2. **Differentiate** to find expressions for \( y'_2 \) and \( y''_2 \).
3. **Simplify** the resulting equation, using the fact that \( e^x \) is a known solution.
4. **Solve** for \( v(x) \).
5. **Integrate** to find the general solution.
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