Question

Find a fundamental set of solutions, given that $y_1$ is a solution. $x^2(\ln |x|{)}^2{y}^{\prime \prime }-(2x\ln |x|)y^{\prime }+(2+\ln |x|)y=0;y_1=\ln |x|$

Short answer

Question: Given the homogeneous second-order linear differential equation $x^2{y}^{″}-x{y}^{\prime }+y=0$ and a solution $y_1=\ln |x|$, find a second linearly independent solution, $y_2$, to form a fundamental set of solutions.

Step-by-step solution

Check the given solution

First, verify that $y_1=\ln |x|$ is a solution to the given differential equation. Plug $y_1$ into the equation and check if it is true.

Reduction of order

With the method of reduction of order, we will assume a solution of the form $y_2=\ln |x|v(x)$, where we will find the function $v(x)$. Plug $y_2$ into the differential equation and its derivatives.

Simplify the equation

After substituting $y_2$ and its derivatives in the differential equation, we will have an equation involving $\ln |x|$ and function $v(x)$ and its derivatives. Simplify the equation by grouping terms with $v^{\prime }(x)$ and $v^{″}(x)$.

Solve for $v^{\prime }(x)$

Observe that the equation obtained in Step 3 is in fact a first-order linear differential equation for $v^{\prime }(x)$. Solve this first-order linear differential equation to obtain $v^{\prime }(x)$.

Get $v(x)$

Integrate $v^{\prime }(x)$ to find the function $v(x)$.

Final solution

Write down the second solution as $y_2=\ln |x|v(x)$. The fundamental set of solutions is composed of $y_1=\ln |x|$ and $y_2=\ln |x|v(x)$.