Exponential Isolation Cc3Bea

1. **State the problem:** We are given the function $$y = C_1 e^x + C_2 e^{2x} + C_3 e^{3x}$$ and asked to express it by isolation, which means solving for one of the constants or terms if possible. 2. **Understanding the function:** This is a linear combination of exponential functions with different exponents. Each term is independent, so isolating one term means expressing it in terms of $$y$$ and the other terms. 3. **Isolate a term, for example, $$C_1 e^x$$:** $$C_1 e^x = y - C_2 e^{2x} - C_3 e^{3x}$$ 4. **Similarly, isolate $$C_2 e^{2x}$$:** $$C_2 e^{2x} = y - C_1 e^x - C_3 e^{3x}$$ 5. **Or isolate $$C_3 e^{3x}$$:** $$C_3 e^{3x} = y - C_1 e^x - C_2 e^{2x}$$ 6. **Summary:** Isolating any term involves subtracting the other terms from $$y$$. This is straightforward because the function is a sum of independent exponential terms. **Final isolated forms:** $$y = C_1 e^x + C_2 e^{2x} + C_3 e^{3x}$$ $$C_1 e^x = y - C_2 e^{2x} - C_3 e^{3x}$$ $$C_2 e^{2x} = y - C_1 e^x - C_3 e^{3x}$$ $$C_3 e^{3x} = y - C_1 e^x - C_2 e^{2x}$$