The homogeneous differential equation 7x2y′′+15xy′+y=07x2y″+15xy′+y=0 has general solution:

y=ax+bx17.y=ax+bx17.

The nonhomogeneous equation 7x2y′′+15xy′+y=96x+47x2y″+15xy′+y=96x+4 has the particular solution:

y=4+6x.y=4+6x.

By superposition, the general solution of the equation 7x2y′′+15xy′+y=96x+47x2y″+15xy′+y=96x+4 is
y=

Given the differential equation:

\[
7x^2y'' + 15xy' + y = 96x + 4
\]

### Step 1: General Solution of the Homogeneous Equation

The homogeneous equation corresponding to the given nonhomogeneous equation is:

\[
7x^2y'' + 15xy' + y = 0
\]

The general solution of the homogeneous equation is given as:

\[
y_h = ax + bx^{\frac{1}{7}}
\]

where \(a\) and \(b\) are arbitrary constants.

### Step 2: Particular Solution of the Nonhomogeneous Equation

The particular solution given for the nonhomogeneous equation is:

\[
y_p = 4 + 6x
\]

### Step 3: General Solution by Superposition

The general solution of the nonhomogeneous differential equation is the sum of the general solution of the homogeneous equation and the particular solution:

\[
y = y_h + y_p
\]
Substituting the expressions:

\[
y = (ax + bx^{\frac{1}{7}}) + (4 + 6x)
\]

Simplify the expression:

\[
y = (a + 6)x + bx^{\frac{1}{7}} + 4
\]

### Final Answer:
\[
y = (a + 6)x + bx^{\frac{1}{7}} + 4
\]

Would you like more details or have any other questions?

Here are five related questions you might find interesting:
1. How do you determine the form of a particular solution for a nonhomogeneous differential equation?
2. Why is the superposition principle applicable to linear differential equations?
3. How would the solution change if the nonhomogeneous term were different, such as \(96x^2 + 4\)?
4. Can you provide a step-by-step derivation of the general solution for the homogeneous equation?
5. How can the method of undetermined coefficients be used to find particular solutions for nonhomogeneous equations?

**Tip:** Always verify the general solution by plugging it back into the original differential equation to ensure it satisfies both the homogeneous and nonhomogeneous parts.

Explore the general and particular solutions of differential equations, specifically focusing on the homogeneous and nonhomogeneous cases. Learn how superposition is applied to derive the general solution from the given differential equation.

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