Question

Asked Jun 19, 2019

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Reduction of order can be used to find the general solution of a non-homogeneous equation.

a_{2}(x)y"+a_{1}(x)y'+a_{0}(x)y=g(x)

whenever a solution y_{1} of the associated homogeneous equation is known. In the case of a general homogeneous equation g(x)=0, it turns out this equation can be reduced to a linear first order differential equation by means of a substitution of a non-trivial solution y_{1}.

Follow the steps below to use the method of reduction of order to find a second solution y_{2} to the following differential equation and y_{1}, which solves the given homogeneous equation.

xy"+y'=0; y_{1}=ln(x)

- Let y
_{2}=uy_{1}, for u=u(x), and find y_{2}' and y_{2}" - Plug y
_{2}' and y_{2}" into the differential equation and simplify - Use w=u' to transform the previous answer into a linear first order differential equation in w.
- Solve for w, and thus u' in the previous answer
- Solve for u from the previous equation for u'.
- Solve for y
_{2}using the definition in step 1 (use c_{1}=-1 and c_{2}=0, the constants of integration) - Verify that y
_{2}is a solution to the original differential equation, and check that W[y_{1},y_{2}] does not equal zero on I=all real numbers

Step 1

Consider the given equation:

Step 2

Now, follow the steps as provided in the question:

Let

Step 3

Now, plug these values in...

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