Solve the differential equation: (1+x²)y'' + 2xy' - $2y = 0$ given that y₁ = x is a known solution.

Step 1: Standard Form and Substitution

First, we rewrite the given differential equation in the standard form $y'' + P(x)y' + Q(x)y = 0$. This allows us to identify $P(x)$ and $Q(x)$. We are given one solution $y_1 = x$, and we will use the method of reduction of order to find a second linearly independent solution $y_2$. We assume $y_2 = v(x)y_1 = vx$.

Step 2: Calculate Derivatives of y₂

Next, we calculate the first and second derivatives of $y_2 = vx$ with respect to $x$. These derivatives will be substituted back into the original differential equation.

Step 3: Substitute into Differential Equation

Substitute $y_2$, $y_2'$, and $y_2''$ into the original differential equation. This step is crucial for transforming the second-order differential equation into a first-order one in terms of $v'$.

Step 4: Simplify and Solve for v'

Expand and simplify the equation. Notice that terms involving $v$ cancel out, which is expected when $y_1$ is a solution. We then rearrange the equation to separate variables for $v''$ and $v'$.

Step 5: Integrate to find v'

Integrate both sides with respect to $x$. We use partial fraction decomposition for the right-hand side integral. This gives us an expression for $v'$.

Step 6: Integrate to find v and y₂

Integrate $v'$ to find $v$. Then, substitute $v$ back into $y_2 = vx$ to find the second linearly independent solution. We can choose $C_2=1$ and $C_3=0$ to get a simple form for $y_2$.

Step 7: General Solution

The general solution is a linear combination of the two linearly independent solutions $y_1$ and $y_2$. We replace $C_1$ and $C_2$ with arbitrary constants $A$ and $B$ for clarity.

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