ODE_Exercises_2010

ODE_Exercises_2010 - Problems in Differential Equations 1....

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Problems in Differential Equations 1. First order differential equations Basic concepts 1. Check that the function y = sin x/x is a solution of the differential equation xy ± + y = cos x. 2. Prove that the function y = Cx 3 is the general solution of the differential equation xy ± - 3 y =0 . Find the particular solution satisfying the condition y (1) = 1 . 3. Find the differential equation of the family of circles x 2 + y 2 =2 ax . Show that the following expression define the general solutions or the gen- eral integrals of the indicated differential equation: 4. y = x ( C - ln | x | ) , ( x - y ) dx + xdy . 5. y = x 0 1 x e x dx + C ) ,x y ± - y = xe x . 6. 2 x + y - 1= Ce 2 - x , (2 x + y - 1) dx - (4 x +2 y - 3) dy . Write the differential equation of the indicated family of curves: 7. Parabol y = x 2 ax. 8. Hyperbol y = a/x . 9. Catenaries y = a cosh x. 1
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2 10. Hyperbol x 2 - y 2 =2 ax . Equations with variables separable 1. Solve the equation dy dx = 2 x 3 y 2 +1 . 2. Solve the equation dy dx = y tan x. Solve the given differential equations: 3. y ± = x/y. 4. y 2 y ± + x 2 =1 . 5. yy ± + x =0 . 6. xy ± y. 7. ( x +1) y ± + xy . 8. y ± 1 - x 2 =1+ y 2 . 9. y ± = e x + y . 10. y ± + x sin x y cos y . 11. (1 + y 2 ) xdx +(1+ y 2 ) dy . 12. xydx + 1 - x 2 dy . 13. ye 2 x dx - (1 + e 2 x ) dy . 14. 2 e x tan ydx e x ) sec 2 ydy . 15. (1 + y )( e x dx - e 2 y dy ) - (1 + y 2 ) dy . 16. (1 + x 2 ) dy + y 1+ x 2 dx - xydx . 17. dy - 2 y ln xdx . 18. y ± = cos( x + y ) .
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3 19. y ± = 1 2 x + y . 20. y ± =(4 x + y +1) 2 . Find the particular solution of the given equation satisfying the indicated initial conditions: 21. y ± = sin( y - x - 1) . 22. y ± +2 y =3 x +5 . 23. y ± = 3 ± (4 x - y 2 . 24. (1 + y 2 ) dx - xydy =0 ,y (1) = 0 . 25. ( xy 2 + x ) dy +( x 2 y - y ) dx (1) = 1 . 26. y ± tg x = y, y ( π/ 2=1) . Homogeneous equations 1. Solve the equation y ± = y x + cos y x . Solve the following equations 2. y ± = y x + x y . 3. y ± = y x + sin y x . 4. y ± =( x - y ) / ( x + y ) . 5. ( x 2 + xy ) y ± = x ± x 2 - y 2 + xy + y 2 . 6. ( x - y ) dx + xdy . 7. y 2 dx + x 2 dy - xydy . 8. x ( y ± + e y/x )= y. 9. xdy - y cos ln y/x . 10. xy ± = y + x tg y/x.
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4 11. xy ± - y = ± x 2 - y 2 . 12. ( x 2 + y 2 ) dy - 2 xydx =0 .
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This note was uploaded on 10/21/2010 for the course A a taught by Professor A during the Spring '10 term at American International.

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ODE_Exercises_2010 - Problems in Differential Equations 1....

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