Find values of A and B so that the function y(x)=Ae×+Bxe×+x²e× satisfy the initial conditions y(1)=1,y'(1)=-1

To find the values of \( A \) and \( B \) such that the function \( y(x) = Ae^x + Bxe^x + x^2e^x \) satisfies the initial conditions \( y(1) = 1 \) and \( y'(1) = -1 \), we need to follow these steps:

1. **Calculate \( y(1) \):**

\[ y(1) = A e^1 + B(1)e^1 + (1)^2e^1 \]
\[ y(1) = A e + B e + e \]
\[ y(1) = e(A + B + 1) \]

Given that \( y(1) = 1 \):

\[ e(A + B + 1) = 1 \]
\[ A + B + 1 = \frac{1}{e} \]
\[ A + B = \frac{1}{e} - 1 \quad \text{(Equation 1)} \]

2. **Calculate \( y'(x) \):**

\[ y(x) = Ae^x + Bxe^x + x^2e^x \]

The derivative is:

\[ y'(x) = A e^x + B(e^x + xe^x) + (2xe^x + x^2e^x) \]
\[ y'(x) = A e^x + B e^x + B x e^x + 2 x e^x + x^2 e^x \]
\[ y'(x) = (A + B) e^x + (B x + 2 x) e^x + x^2 e^x \]
\[ y'(x) = (A + B + (B + 2)x + x^2) e^x \]

3. **Calculate \( y'(1) \):**

\[ y'(1) = (A + B + (B + 2)(1) + (1)^2) e \]
\[ y'(1) = (A + B + B + 2 + 1) e \]
\[ y'(1) = (A + 2B + 3) e \]

Given that \( y'(1) = -1 \):

\[ (A + 2B + 3) e = -1 \]
\[ A + 2B + 3 = -\frac{1}{e} \quad \text{(Equation 2)} \]

Now we have two equations:

1. \( A + B = \frac{1}{e} - 1 \)
2. \( A + 2B + 3 = -\frac{1}{e} \)

We can solve these simultaneously.

From Equation 1:

\[ A = \frac{1}{e} - 1 - B \]

Substitute \( A \) in Equation 2:

\[ \left( \frac{1}{e} - 1 - B \right) + 2B + 3 = -\frac{1}{e} \]
\[ \frac{1}{e} - 1 - B + 2B + 3 = -\frac{1}{e} \]
\[ \frac{1}{e} + B + 2 = -\frac{1}{e} \]
\[ B + 2 = -\frac{2}{e} \]
\[ B = -\frac{2}{e} - 2 \]

Substitute \( B \) back into \( A + B = \frac{1}{e} - 1 \):

\[ A + \left( -\frac{2}{e} - 2 \right) = \frac{1}{e} - 1 \]
\[ A - \frac{2}{e} - 2 = \frac{1}{e} - 1 \]
\[ A = \frac{1}{e} - 1 + \frac{2}{e} + 

Learn how to find values of A and B for the function y(x) = Ae^x + Bxe^x + x^2e^x that satisfy the initial conditions y(1) = 1 and y'(1) = -1. This problem involves advanced concepts in exponential functions, derivatives, and solving initial value problems. Suitable for advanced high school students or undergraduate mathematics.

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