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У (8х — 9у)dx + 2x (х — Зу)dy %3D 0 а. 2x4у — Зху? — С b. 2x*y – 3x²y? = C c. 2x*y³ – 3xy² = C d. None of the above

У (8х — 9у)dx + 2x (х — Зу)dy %3D 0 а. 2x4у — Зху? — С b. 2x*y – 3x²y? = C c. 2x*y³ – 3xy² = C d. None of the above

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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У (8х — 9у)dx + 2x (х — Зу)dy %3D 0 а. 2x4у — Зху? — С b. 2x*y – 3x²y? = C c. 2x*y³ – 3xy² = C d. None of the above
Transcribed Image Text:У (8х — 9у)dx + 2x (х — Зу)dy %3D 0 а. 2x4у — Зху? — С b. 2x*y – 3x²y? = C c. 2x*y³ – 3xy² = C d. None of the above
Step i The given differential equation is: 2(2y? + 5ху — 2у+ 4) dx + x(2х + 2у — 1) dy %3D 0 The above differential equation can be rewritten as, (4у? + 10ху — 4у +8)dx + (2x? + 2ху — х)dy %3D 0 (4y dx + 2xydy) + (10xydx + 2x dy) – (4ydx + xdy) + (8dx) = 0 - Step 2 Now if we multiply with x on both sides, then the differential equation is: (4x³y²dx + 2x*ydy) – (4x*ydx + x*dy) + (8x*dx) + (10x“ydx + 2x° dy) = 0 Clearly, 4x³ y² dx + 2x*ydy = d(x*y²) (4x³ ydx + x*dy) = d(x^y) (8x*dx) = d (2x*) (10x*ydx + 2x°dy) = d(2x°y) Therefore, |(4x'y²dx + 2x*ydy) – (4x³ydx +x*dy) + (8x*dx) + (10x*ydx + 2r°dy) = 0 |d(x*y²) – d(x*y) + d (2x+) + d(2x°y) = 0 - Now integrating on both sides, S d(x*y²) – S d(x*y) + S d (2x*) + [ d(2x°y) = / 0 x*y² – x*y+ 2x+ + 2x°y = C x*y? + 2x*y+ 2x* – x*y = C
Transcribed Image Text:Step i The given differential equation is: 2(2y? + 5ху — 2у+ 4) dx + x(2х + 2у — 1) dy %3D 0 The above differential equation can be rewritten as, (4у? + 10ху — 4у +8)dx + (2x? + 2ху — х)dy %3D 0 (4y dx + 2xydy) + (10xydx + 2x dy) – (4ydx + xdy) + (8dx) = 0 - Step 2 Now if we multiply with x on both sides, then the differential equation is: (4x³y²dx + 2x*ydy) – (4x*ydx + x*dy) + (8x*dx) + (10x“ydx + 2x° dy) = 0 Clearly, 4x³ y² dx + 2x*ydy = d(x*y²) (4x³ ydx + x*dy) = d(x^y) (8x*dx) = d (2x*) (10x*ydx + 2x°dy) = d(2x°y) Therefore, |(4x'y²dx + 2x*ydy) – (4x³ydx +x*dy) + (8x*dx) + (10x*ydx + 2r°dy) = 0 |d(x*y²) – d(x*y) + d (2x+) + d(2x°y) = 0 - Now integrating on both sides, S d(x*y²) – S d(x*y) + S d (2x*) + [ d(2x°y) = / 0 x*y² – x*y+ 2x+ + 2x°y = C x*y? + 2x*y+ 2x* – x*y = C
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