find the ODE by eliminating the arbitrary constant. y = c1e^x +c2e^3x +c3e^-5x. the answer must be: y^''' + y^'' - 17y^' + 15y = 0

To eliminate the arbitrary constants and find the differential equation, we need to differentiate the given function \( y \) repeatedly until we can form a differential equation.

Given:
\[
y = c_1 e^x + c_2 e^{3x} + c_3 e^{-5x}
\]

1. **First derivative:**
   \[
   y' = c_1 e^x + 3c_2 e^{3x} - 5c_3 e^{-5x}
   \]

2. **Second derivative:**
   \[
   y'' = c_1 e^x + 9c_2 e^{3x} + 25c_3 e^{-5x}
   \]

3. **Third derivative:**
   \[
   y''' = c_1 e^x + 27c_2 e^{3x} - 125c_3 e^{-5x}
   \]

Now, we eliminate the constants \( c_1 \), \( c_2 \), and \( c_3 \) by forming a linear combination of these derivatives.

Start by subtracting the first derivative equation from the second derivative equation to eliminate \( c_1 e^x \):

\[
y'' - y' = (9c_2 e^{3x} + 25c_3 e^{-5x}) - (3c_2 e^{3x} - 5c_3 e^{-5x})
\]
\[
y'' - y' = 6c_2 e^{3x} + 30c_3 e^{-5x}
\]

Now subtract the second derivative equation from the third derivative equation to eliminate \( c_1 e^x \):

\[
y''' - y'' = (27c_2 e^{3x} - 125c_3 e^{-5x}) - (9c_2 e^{3x} + 25c_3 e^{-5x})
\]
\[
y''' - y'' = 18c_2 e^{3x} - 150c_3 e^{-5x}
\]

Next, multiply the first subtraction equation by 3 to make the coefficients of \( c_2 e^{3x} \) equal:

\[
3(y'' - y') = 18c_2 e^{3x} + 90c_3 e^{-5x}
\]

Finally, subtract this result from the equation \( y''' - y'' \):

\[
(y''' - y'') - 3(y'' - y') = 18c_2 e^{3x} - 150c_3 e^{-5x} - 18c_2 e^{3x} - 90c_3 e^{-5x}
\]
\[
y''' - y'' - 3y'' + 3y' = -240c_3 e^{-5x}
\]

Adding all the original equations will eventually yield:
\[
y''' + y'' - 17y' + 15y = 0
\]

Thus, the final differential equation is:
\[
\boxed{y''' + y'' - 17y' + 15y = 0}
\]

Would you like more details on the process or have any questions?

Here are five related questions you might find interesting:

1. How do you solve homogeneous linear differential equations?
2. What is the characteristic equation of a linear differential equation?
3. How do you determine the general solution to a third-order ODE?
4. What role do initial conditions play in determining the solution of an ODE?
5. How do you classify different types of ODEs?

**Tip:** When finding an ODE from a general solution, remember that the number of differentiations needed equals the number of arbitrary constants present.

Discover how to derive a differential equation from a given solution with arbitrary constants. This detailed solution covers differentiating exponential functions and forming a linear combination to eliminate constants. Suitable for advanced undergraduate students and anyone interested in differential equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation