The differential equation provided in the image is:

$\frac{dy}{dx} = 7x^{13} + 10 - 5e^x$

Part A: General Solution

To solve the differential equation, we integrate both sides with respect to $x$:

$y = \int \left( 7x^{13} + 10 - 5e^x \right) dx$

Breaking it down:

• $\int 7x^{13} dx = \frac{7x^{14}}{14} = \frac{x^{14}}{2}$
• $\int 10 dx = 10x$
• $\int -5e^x dx = -5e^x$

Therefore, the general solution is:

$y = \frac{x^{14}}{2} + 10x - 5e^x + C$

where $C$ is the constant of integration.

Part B: Initial Value Problem (IVP)

We are given that the solution curve passes through the point $(0, 37)$, meaning when $x = 0$, $y = 37$. Substituting $x = 0$ and $y = 37$ into the general solution:

$37 = \frac{0^{14}}{2} + 10(0) - 5e^0 + C$ $37 = 0 + 0 - 5 + C$ $C = 37 + 5 = 42$

Thus, the particular solution is:

$y = \frac{x^{14}}{2} + 10x - 5e^x + 42$

Would you like more details or further clarifications? Here are 5 related questions you might explore:

1. How can we verify that this solution satisfies the original differential equation?
2. What would happen if the differential equation included a different initial condition?
3. Can this solution be expressed in terms of another variable substitution?
4. How would the solution change if the function was non-homogeneous?
5. What is the importance of the constant $C$ in the general solution?

Tip: Always remember to apply the initial condition carefully to find the correct value of $C$ when solving an IVP.