Pregunta
6. Given that \( \frac{d y}{d x}=2 x+7 \) and \( y=5 \) when \( x=-1 \), find the value of \( y \) when \( x=2 \).
Ask by Lyons Kelley. in Malaysia
Jan 21,2025
Solución de tutoría real
Respuesta verificada por el tutor
Responder
When \( x = 2 \), \( y = 29 \).
Solución
To find the value of \( y \) when \( x = 2 \), follow these steps:
1. **Given Differential Equation:**
\[
\frac{dy}{dx} = 2x + 7
\]
2. **Integrate to Find \( y \):**
\[
y = \int (2x + 7) \, dx = x^2 + 7x + C
\]
where \( C \) is the constant of integration.
3. **Apply the Initial Condition \( y = 5 \) when \( x = -1 \):**
\[
5 = (-1)^2 + 7(-1) + C \\
5 = 1 - 7 + C \\
5 = -6 + C \\
C = 11
\]
4. **Write the Particular Solution:**
\[
y = x^2 + 7x + 11
\]
5. **Find \( y \) When \( x = 2 \):**
\[
y = (2)^2 + 7(2) + 11 = 4 + 14 + 11 = 29
\]
**Answer:** \( y = 29 \) when \( x = 2 \).
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución Mind Expander
To find the value of \( y \) when \( x=2 \), we start by integrating the given derivative \( \frac{d y}{d x}=2 x+7 \). 1. Integrate the equation: \[ y = \int (2x + 7) \, dx = x^2 + 7x + C \] where \( C \) is the constant of integration. 2. We know \( y = 5 \) when \( x = -1 \). We can use this to find \( C \): \[ 5 = (-1)^2 + 7(-1) + C \] \[ 5 = 1 - 7 + C \implies C = 5 + 6 = 11 \] 3. Now we have the equation: \[ y = x^2 + 7x + 11 \] 4. To find \( y \) when \( x = 2 \): \[ y = (2)^2 + 7(2) + 11 \] \[ y = 4 + 14 + 11 = 29 \] Thus, the value of \( y \) when \( x = 2 \) is \( \boxed{29} \).
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