Compute y' and y". The symbols C₁ and C₂ represent constants. y = C₁e* + C₂xex y'(x) = y"(x) = Combine these derivatives with y as a linear second-order differential equation that is free of the symbols C₁ and C₂ and has the form F(y, y', y") = 0. (Use yp fo = 0
Compute y' and y". The symbols C₁ and C₂ represent constants. y = C₁e* + C₂xex y'(x) = y"(x) = Combine these derivatives with y as a linear second-order differential equation that is free of the symbols C₁ and C₂ and has the form F(y, y', y") = 0. (Use yp fo = 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![### Derivatives of a Given Function
To solve the following differential equations for educational purposes, please follow along with the steps outlined below.
**Given Function:**
\[ y = C_1 e^x + C_2 x e^x \]
where \( C_1 \) and \( C_2 \) represent constants.
**First Derivative:**
To find the first derivative, \( y'(x) \):
\[ y'(x) = \frac{d}{dx} \left(C_1 e^x + C_2 x e^x\right) \]
Using the product rule and the derivative of \( e^x \):
\[ y'(x) = C_1 e^x + C_2 \left(e^x + x e^x\right) \]
\[ y'(x) = C_1 e^x + C_2 e^x + C_2 x e^x \]
\[ y'(x) = (C_1 + C_2) e^x + C_2 x e^x \]
**Second Derivative:**
To find the second derivative, \( y''(x) \):
\[ y''(x) = \frac{d}{dx} \left( (C_1 + C_2) e^x + C_2 x e^x \right) \]
Using the product rule again:
\[ y''(x) = (C_1 + C_2) e^x + C_2 \left(e^x + x e^x\right) \]
\[ y''(x) = (C_1 + C_2) e^x + C_2 e^x + C_2 x e^x \]
\[ y''(x) = (C_1 + 2C_2) e^x + C_2 x e^x \]
**Combining Derivatives into a Linear Second-Order Differential Equation:**
We need to combine these derivatives with \( y \) as a linear second-order differential equation that is free of the symbols \( C_1 \) and \( C_2 \) and has the form \( F(y, y', y'') = 0 \):
\[ y'' - 2y' + y = 0 \]
This step ensures the resulting equation is homogeneous and does not include the constants \( C_1 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0436118d-47b7-4fa9-abd3-dac72bbeccd1%2Fbf254329-bbdf-4fba-b7f0-037f0a6ca484%2Fmadgdrj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Derivatives of a Given Function
To solve the following differential equations for educational purposes, please follow along with the steps outlined below.
**Given Function:**
\[ y = C_1 e^x + C_2 x e^x \]
where \( C_1 \) and \( C_2 \) represent constants.
**First Derivative:**
To find the first derivative, \( y'(x) \):
\[ y'(x) = \frac{d}{dx} \left(C_1 e^x + C_2 x e^x\right) \]
Using the product rule and the derivative of \( e^x \):
\[ y'(x) = C_1 e^x + C_2 \left(e^x + x e^x\right) \]
\[ y'(x) = C_1 e^x + C_2 e^x + C_2 x e^x \]
\[ y'(x) = (C_1 + C_2) e^x + C_2 x e^x \]
**Second Derivative:**
To find the second derivative, \( y''(x) \):
\[ y''(x) = \frac{d}{dx} \left( (C_1 + C_2) e^x + C_2 x e^x \right) \]
Using the product rule again:
\[ y''(x) = (C_1 + C_2) e^x + C_2 \left(e^x + x e^x\right) \]
\[ y''(x) = (C_1 + C_2) e^x + C_2 e^x + C_2 x e^x \]
\[ y''(x) = (C_1 + 2C_2) e^x + C_2 x e^x \]
**Combining Derivatives into a Linear Second-Order Differential Equation:**
We need to combine these derivatives with \( y \) as a linear second-order differential equation that is free of the symbols \( C_1 \) and \( C_2 \) and has the form \( F(y, y', y'') = 0 \):
\[ y'' - 2y' + y = 0 \]
This step ensures the resulting equation is homogeneous and does not include the constants \( C_1 \
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