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Solve the differential equation (2x³ − 8xy³ – 18x) dx + (12x²y² + 3y³ − 6y³) dy = 0 using the following steps. (i) Show that by applying the substitution u = x², v = y³, the differential equation becomes M(u, v) du + N(u, v) dy = 0, where M and N are linear functions of u and v. (ii) Find the unique solution (uo, vo) of the linear system of equations JM(u, v) = 0 N(u, v) = 0. Afterwards, apply another substitution r = u-uo, s = v-vo. The new differential equation in r and s is now homogeneous and can be solved using the method you learned in class.

Solve the differential equation (2x³ − 8xy³ – 18x) dx + (12x²y² + 3y³ − 6y³) dy = 0 using the following steps. (i) Show that by applying the substitution u = x², v = y³, the differential equation becomes M(u, v) du + N(u, v) dy = 0, where M and N are linear functions of u and v. (ii) Find the unique solution (uo, vo) of the linear system of equations JM(u, v) = 0 N(u, v) = 0. Afterwards, apply another substitution r = u-uo, s = v-vo. The new differential equation in r and s is now homogeneous and can be solved using the method you learned in class.

Advanced Engineering Mathematics
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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Solve the differential equation (2x³ − 8xy³ – 18x) dx + (12x²y² + 3y³ − 6y³) dy = 0 using the following steps. (i) Show that by applying the substitution u = x², v = y³, the differential equation becomes M(u, v) du + N(u, v) dy = 0, where M and N are linear functions of u and v. (ii) Find the unique solution (uo, vo) of the linear system of equations JM(u, v) = 0 N(u, v) = 0. Afterwards, apply another substitution r = u-uo, s = v-vo. The new differential equation in r and s is now homogeneous and can be solved using the method you learned in class.
Transcribed Image Text:Solve the differential equation (2x³ − 8xy³ – 18x) dx + (12x²y² + 3y³ − 6y³) dy = 0 using the following steps. (i) Show that by applying the substitution u = x², v = y³, the differential equation becomes M(u, v) du + N(u, v) dy = 0, where M and N are linear functions of u and v. (ii) Find the unique solution (uo, vo) of the linear system of equations JM(u, v) = 0 N(u, v) = 0. Afterwards, apply another substitution r = u-uo, s = v-vo. The new differential equation in r and s is now homogeneous and can be solved using the method you learned in class.
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