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18 through 21, find an integrating factor and solve In each of Problems the given equation. 18. (3x²y + 2xy + y³) + (x² + y²) y = 0 19. y' = e2x + y - 1 20. 1+(x/y-sin y) y' = 0 21. y+ (2xy-e-2y) y' = 0 PCC tial equation

18 through 21, find an integrating factor and solve In each of Problems the given equation. 18. (3x²y + 2xy + y³) + (x² + y²) y = 0 19. y' = e2x + y - 1 20. 1+(x/y-sin y) y' = 0 21. y+ (2xy-e-2y) y' = 0 PCC tial equation

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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19,21

 

2. 3. 4. 5. Substituting this expression for (x, y) in the second of equations (32), we find that x³ + x²y+h'(y) = x³ + x²y, so h'(y) = 0 and h(y) is a constant. Thus the solutions of equation (31), and hence of equation (19), are given implicitly by Problems Determine whether each of the equations in Problems 1 through 8 is 14. Show that any separable equation exact. If it is exact, find the solution. M(x) + N(y) y = 0 1. (2x+3)+(2y-2)y' = 0 6. 7. 8. Solutions may also be found in explicit form since equation (33) is quadratic in y. You may also verify that a second integrating factor for equation (19) is 1 μ(x, y) xy(2x + y) and that the same solution is obtained, though with much greater difficulty, if this integrating factor is used (see Problem 22). (2x+4y) + (2x - 2y) y' = 0 (3x² - 2xy + 2) + (6y²-x²+3) y' = 0 dy ax + by dx bx + cy ax - by bx - cy dy dx x³y + ²x²y² = c C. = (yety cos(2x) -2ey sin(2x)+2x)+(xey cos(2x) - 3) y' = 0 (y/x+6x) + (lnx - 2) y' = 0, x > 0 X + y dy =0 (x² + y2)3/2 (x² + y²)3/2 dx In each of Problems 9 and 10, solve the given initial value problem and determine at least approximately where the solution is valid. y(1) = 3 9. (2x - y) + (2y-x) y' = 0, 10. (9x² + y - 1) - (4y - x) y' = 0, y(1) = 0 In each of Problems 11 and 12, find the value of b for which the given equation is exact, and then solve it using that value of b. 11. (xy² + bx²y) + (x + y)x²y' = 0 12. (ye²xy + x) + bxe²xy y' = 0 13. Assume that equation (6) meets the requirements of Theorem 2.6.1 in a rectangle R and is therefore exact. Show that a possible function(x, y) is $(x, y) = f where (xo, yo) is a point in R. M(s, yo) ds + s+ S₁0^ (33) N(x, t) dt, is also exact. In each of Problems 15 and 16, show that the given equation is not exact but becomes exact when multiplied by the given integrating factor. Then solve the equation. 15. x²y³ + x(1+ y²) y' = 0, u(x, y) = 1/(xy³) 16. (x + 2) sin y + (x cos y) y' = 0, u(x, y) = xe* 17. Show that if (Nx - My)/M = Q, where Q is a function of y only, then the differential equation M + Ny' = 0 has an integrating factor of the form p/ μ(y) = exp Q(y)dy. In each of Problems 18 through 21, find an integrating factor and solve the given equation. 18. (3x²y + 2xy + y³) + (x² + y²) y' = 0 19. y' = e²x + y - 1 20. 1+ (x/y-sin y) y' = 0 21. y + (2xy-e-²y) y' = 0 22. Solve the differential equation (3xy + y²) + (x² + xy) y' = 0 using the integrating factor u(x, y) = (xy(2x + y))-¹. Verify that the solution is the same as that obtained in Example 4 with a different integrating factor.
Transcribed Image Text:2. 3. 4. 5. Substituting this expression for (x, y) in the second of equations (32), we find that x³ + x²y+h'(y) = x³ + x²y, so h'(y) = 0 and h(y) is a constant. Thus the solutions of equation (31), and hence of equation (19), are given implicitly by Problems Determine whether each of the equations in Problems 1 through 8 is 14. Show that any separable equation exact. If it is exact, find the solution. M(x) + N(y) y = 0 1. (2x+3)+(2y-2)y' = 0 6. 7. 8. Solutions may also be found in explicit form since equation (33) is quadratic in y. You may also verify that a second integrating factor for equation (19) is 1 μ(x, y) xy(2x + y) and that the same solution is obtained, though with much greater difficulty, if this integrating factor is used (see Problem 22). (2x+4y) + (2x - 2y) y' = 0 (3x² - 2xy + 2) + (6y²-x²+3) y' = 0 dy ax + by dx bx + cy ax - by bx - cy dy dx x³y + ²x²y² = c C. = (yety cos(2x) -2ey sin(2x)+2x)+(xey cos(2x) - 3) y' = 0 (y/x+6x) + (lnx - 2) y' = 0, x > 0 X + y dy =0 (x² + y2)3/2 (x² + y²)3/2 dx In each of Problems 9 and 10, solve the given initial value problem and determine at least approximately where the solution is valid. y(1) = 3 9. (2x - y) + (2y-x) y' = 0, 10. (9x² + y - 1) - (4y - x) y' = 0, y(1) = 0 In each of Problems 11 and 12, find the value of b for which the given equation is exact, and then solve it using that value of b. 11. (xy² + bx²y) + (x + y)x²y' = 0 12. (ye²xy + x) + bxe²xy y' = 0 13. Assume that equation (6) meets the requirements of Theorem 2.6.1 in a rectangle R and is therefore exact. Show that a possible function(x, y) is $(x, y) = f where (xo, yo) is a point in R. M(s, yo) ds + s+ S₁0^ (33) N(x, t) dt, is also exact. In each of Problems 15 and 16, show that the given equation is not exact but becomes exact when multiplied by the given integrating factor. Then solve the equation. 15. x²y³ + x(1+ y²) y' = 0, u(x, y) = 1/(xy³) 16. (x + 2) sin y + (x cos y) y' = 0, u(x, y) = xe* 17. Show that if (Nx - My)/M = Q, where Q is a function of y only, then the differential equation M + Ny' = 0 has an integrating factor of the form p/ μ(y) = exp Q(y)dy. In each of Problems 18 through 21, find an integrating factor and solve the given equation. 18. (3x²y + 2xy + y³) + (x² + y²) y' = 0 19. y' = e²x + y - 1 20. 1+ (x/y-sin y) y' = 0 21. y + (2xy-e-²y) y' = 0 22. Solve the differential equation (3xy + y²) + (x² + xy) y' = 0 using the integrating factor u(x, y) = (xy(2x + y))-¹. Verify that the solution is the same as that obtained in Example 4 with a different integrating factor.
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