Reduction of order can be used to find the general solution of a non-homogeneous equation.a2(x)y"+a1(x)y'+a0(x)y=g(x)whenever a solution y1 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 y1.Follow the steps below to use the method of reduction of order to find a second solution y2 to the following differential equation and y1, which solves the given homogeneous equation.xy"+y'=0; y1=ln(x)Let y2=uy1, for u=u(x), and find y2' and y2"Plug y2' and y2" into the differential equation and simplifyUse 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 answerSolve for u from the previous equation for u'.Solve for y2 using the definition in step 1 (use c1=-1 and c2=0, the constants of integration)Verify that y2 is a solution to the original differential equation, and check that W[y1,y2] does not equal zero on I=all real numbers

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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

a₂(x)y"+a₁(x)y'+a₀(x)y=g(x)

whenever a solution y₁ 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 y₁.

Follow the steps below to use the method of reduction of order to find a second solution y₂ to the following differential equation and y₁, which solves the given homogeneous equation.

xy"+y'=0; y₁=ln(x)

Let y₂=uy₁, for u=u(x), and find y₂' and y₂"
Plug y₂' and y₂" 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 y₂ using the definition in step 1 (use c₁=-1 and c₂=0, the constants of integration)
Verify that y₂ is a solution to the original differential equation, and check that W[y₁,y₂] does not equal zero on I=all real numbers

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Advanced Engineering Mathematics

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ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated