
2 d
2
y dy
a) Show that y = x is a solution of the equation, 3 
dx2
+ 5 = 10x + 6
dx
b) Check whether y = x
2
+ c where c is any constant is also a solution of the equation.

Let's break down the problem and solve it step by step.

### Given Differential Equation:
\[ \frac{d^2y}{dx^2} + 5\frac{dy}{dx} = 10x + 6 \]

#### Part (a): Show that \( y = x \) is a solution of the equation

1. **First Derivative**: 
   Since \( y = x \), differentiate with respect to \( x \):
   \[
   \frac{dy}{dx} = 1
   \]

2. **Second Derivative**:
   Differentiate \( \frac{dy}{dx} = 1 \) with respect to \( x \):
   \[
   \frac{d^2y}{dx^2} = 0
   \]

3. **Substitute into the given equation**:
   Substitute \( \frac{d^2y}{dx^2} = 0 \) and \( \frac{dy}{dx} = 1 \) into the original equation:
   \[
   0 + 5(1) = 10x + 6
   \]
   Simplifying:
   \[
   5 = 10x + 6
   \]
   This equation does not hold for all values of \( x \), so **\( y = x \) is not a solution** to the differential equation.

#### Part (b): Check whether \( y = x^2 + c \) where \( c \) is any constant is a solution

1. **First Derivative**: 
   For \( y = x^2 + c \):
   \[
   \frac{dy}{dx} = 2x
   \]

2. **Second Derivative**:
   Differentiate \( \frac{dy}{dx} = 2x \) with respect to \( x \):
   \[
   \frac{d^2y}{dx^2} = 2
   \]

3. **Substitute into the given equation**:
   Substitute \( \frac{d^2y}{dx^2} = 2 \) and \( \frac{dy}{dx} = 2x \) into the original equation:
   \[
   2 + 5(2x) = 10x + 6
   \]
   Simplifying:
   \[
   2 + 10x = 10x + 6
   \]
   \[
   2 = 6
   \]
   This equation is also not true for all values of \( x \), so **\( y = x^2 + c \) is not a solution** to the differential equation.

### Summary:
- \( y = x \) is **not** a solution of the differential equation.
- \( y = x^2 + c \) is **not** a solution of the differential equation.

Would you like more details on any specific part of the solution, or do you have any other questions?

### Related Questions:
1. How do you solve second-order linear differential equations?
2. What are the methods to verify a solution to a differential equation?
3. How does the presence of a constant \( c \) affect the solution to a differential equation?
4. Can a particular solution be found for this differential equation?
5. What are some examples of homogeneous differential equations?

### Tip:
When solving differential equations, always verify your potential solution by substituting it back into the original equation to ensure it satisfies all terms.

Explore how to solve second-order linear differential equations with detailed steps. This problem involves verifying specific solutions to a differential equation. Suitable for advanced high school students.

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