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solve for each equation without any integrals

To solve each equation without the integrals, we need to integrate the expressions and then apply any initial conditions if they are provided. Here are the steps and solutions for each equation based on the given forms.

### 1. **Equation for \( S(t) \)**:
\[
S(t) = Ce^{\int (0.789 + \frac{0.127}{N} S(t) I(t)) \, dt}
\]

This equation involves solving a differential equation. Since it's a first-order differential equation, solving for \( S(t) \) requires substituting any initial conditions.

### 2. **Equation for \( E(t) \)**:
\[
E(t) = e^{-0.167t} \cdot e^{\int e^{0.167t} \cdot \frac{0.127 \times S(t) I(t)}{N} \, dt + C}
\]

This can be simplified to:
\[
E(t) = e^{\int \frac{0.127 \times S(t) I(t)}{N} \, dt - 0.167t + C}
\]

### 3. **Equation for \( I(t) \)**:
\[
I(t) = e^{-0.150t} \left( 0.167 \int e^{0.150t} E(t) \, dt + C \right)
\]

This equation shows the relationship between \( I(t) \) and \( E(t) \). The integral of \( E(t) \) needs to be solved, and then multiplied by the exponential decay factor.

### 4. **Equation for \( R(t) \)**:
\[
R(t) = 0.125 \int I(t) \, dt + C
\]

To solve for \( R(t) \), integrate \( I(t) \) with respect to \( t \) and then multiply by 0.125, adding a constant \( C \).

### 5. **Equation for \( V(t) \)**:
\[
V(t) = 0.789 \int S(t) \, dt + C
\]

Integrate \( S(t) \) with respect to \( t \), then multiply by 0.789 and add the constant \( C \).

### 6. **Equation for \( D(t) \)**:
\[
D(t) = 0.025 \int I(t) \, dt + C
\]

This is similar to \( R(t) \) but with a different coefficient. Integrate \( I(t) \) with respect to \( t \), then multiply by 0.025 and add \( C \).

Since the original equations are in terms of integrals, without specific initial conditions or further simplifications, the solutions will remain in their integral form. To fully solve these equations, specific functions for \( S(t) \), \( I(t) \), and \( E(t) \) are needed, which are typically determined by the context or initial conditions provided in a real problem.

### Relative Questions:
1. How do initial conditions affect the solutions of these differential equations?
2. What methods can be used to solve these types of integral differential equations?
3. How does the constant \( C \) play a role in the general solution of these equations?
4. What is the significance of the coefficients like 0.125, 0.789, and 0.025 in the context of these equations?
5. How would the solutions change if the integral expressions for \( S(t) \), \( I(t) \), or \( E(t) \) were explicitly defined?

### Tip:
When solving differential equations involving integrals, always consider any initial conditions that might be provided to simplify and find the exact solution.

Learn how to solve differential equations without integrals using first-order differential equation solutions. This problem involves finding solutions for equations describing the relationships between S(t), E(t), I(t), R(t), V(t), and D(t). Suitable for college-level students studying differential equations.

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