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**Problem 4:**
\[
\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = 4 \sinh x
\]

**Problem 5:**
\[
\frac{d^2y}{dx^2} + \frac{dy}{dx} - 2y = 2 \cosh 2x
\]

I will proceed with solving these equations step by step.

### Problem 4
The equation is a second-order linear non-homogeneous differential equation:
\[
\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = 4 \sinh x
\]
The solution consists of two parts: the complementary (or homogeneous) solution \(y_c\) and the particular solution \(y_p\).

### Problem 5
Similarly, this is another second-order linear non-homogeneous differential equation:
\[
\frac{d^2y}{dx^2} + \frac{dy}{dx} - 2y = 2 \cosh 2x
\]
We'll also find the complementary and particular solutions for this.

I'll solve these equations.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can solve these second-order differential equations using the method of undetermined coefficients or variation of parameters, depending on the non-homogeneous part of the equations.

Would you like me to guide you step-by-step on how to solve them manually, or answer any specific questions?

Solve number 4 and 5 from the given list of differential equations.

Learn how to solve the non-homogeneous second-order differential equations d²y/dx² + 2dy/dx + y = 4 sinh(x) and d²y/dx² + dy/dx - 2y = 2 cosh(2x) using complementary and particular solutions. This tutorial covers key techniques such as undetermined coefficients and variation of parameters, ideal for advanced high school or undergraduate students.

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