Find the solution of the ordinary differential equationdydx=2y-x+52x-y-4

Answered: dy_2y-x+5 dx 2x- y-4

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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$\begin{array}{l} Find the solution of the ordinary differential equation \\ \frac{d y}{d x} = \frac{2 y - x + 5}{2 x - y - 4} \end{array}$

Step 2

$\begin{array}{l} Consider a given differential equation is \\ \frac{d y}{d x} = \frac{2 y - x + 5}{2 x - y - 4} . . . . . . . (1) \\ Put x = X + h and y = Y + k and \frac{d Y}{d X} = \frac{d y}{d x} \\ The equation (1) becomes \\ \frac{d Y}{d X} = \frac{2 (Y + k) - (X + h) + 5}{2 (X + h) - (Y + k) - 4} \\ \frac{d Y}{d X} = \frac{2 Y - X + 2 k - h + 5}{2 X - Y + 2 h - k - 4} . . . . . . (2) \\ Put 2 k - h + 5 = 0 and 2 h - k - 4 = 0 . . . . . . (3) \\ Hence, equation (2) becomes \\ \frac{d Y}{d X} = \frac{2 Y - X}{2 X - Y} . . . . . . (4) \end{array}$

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