Solve the differential equation: (1+x²)y'' + 2xy' - 2y = 0, given that y₁ = x is a known solution.
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Step-by-Step 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 y1=x, and we will use the method of reduction of order to find a second linearly independent solution y2. We assume y2=v(x)y1=vx.
Step 2: Calculate Derivatives of y₂
Next, we calculate the first and second derivatives of y2=vx with respect to x. These derivatives will be substituted back into the original differential equation.
Step 3: Substitute into Differential Equation
Substitute y2, y2′, and y2′′ 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 y1 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 y2=vx to find the second linearly independent solution. We can choose C2=1 and C3=0 to get a simple form for y2.
Step 7: General Solution
The general solution is a linear combination of the two linearly independent solutions y1 and y2. We replace C1 and C2 with arbitrary constants A and B for clarity.