2.6E: Substitution (Exercises)

 In Exercises 1-6, solve the Bernoulli equation.

1. \(y'+y=y^2\)

2. \( {7xy'-2y=-{x^2 \over y^6}}\)

3. \(x^2y'+2y=2e^{1/x}y^{1/2}\)

4. \( {(1+x^2)y'+2xy ={1 \over (1+x^2)y}}\)

5. \(y'-xy=x^3y^3\)

6. \( {y'-{1+x\over 3x}y=y^4}\)

In Exercises 7-13, solve the initial value problem where the differential equation is a Bernoulli equation.

7. \(y'-2y=xy^3,\quad y(0)=2\sqrt2\)

8. \(y'-xy=xy^{3/2},\quad y(1)=4\)

9. \(xy'+y=x^4y^4,\quad y(1)=1/2\)

10. \(y'-2y=2y^{1/2},\quad y(0)=1\)

11. \( {y'-4y={48x\over y^2},\quad y(0)=1}\)

12. \(x^2y'+2xy=y^3,\quad y(1)=1/\sqrt2\)

13. \(y'-y=xy^{1/2},\quad y(0)=4\)

In Exercises 14-19, show that the differential equation is homogeneous of the same degree. Then solve it using an appropriate substitution.

14. \((y+x)dx-xdy=0\)

15. \((y^2+2xy)dx-x^2dy=0\)

16. \((y^4+x^4)dx-xy^3dy=0\)

17. \(\Big(y+x\sec\Big(\dfrac{y}{x}\Big)\Big) dx-x dy=0\)

18. \((xy+x^2+y^2)dx-x^2dy=0\)

19. \((x^2+2y^2)dx-xydy=0\)

In Exercises 20-25, solve the initial value problem where the differential equation is homogeneous.

20. \((xy+y^2)dx-x^2dy=0, \quad y(-1)=2\)

21. \((x^3+y^3)dx- xy^2dy=0, \quad y(1)=3\)

22. \((x^2+y^2)dx+xydy=0, \quad y(1)=2\)

23. \((y^2-3xy-5x^2)dx-x^2dy=0, \quad y(1)=-1\)

24. \((2x^2+y^2+4xy)dx-x^2dy=0, \quad y(1)=1\)

25. \((3x^2+4y^2)dx-xydy=0, \quad y(1)=\sqrt{3}\)

In Exercises 26-31, solve the differential equation using the indicated substitution.

26. \(\dfrac{dy}{dx}= (x+y+1)^2\quad \text{using}\quad u=x+y+1\)

27. \(\dfrac{dy}{dx}= \tan^2(x+y)\quad \text{using}\quad u=x+y\)

28.  \(\dfrac{dy}{dx}= 2+\sqrt{y-2x+3}\quad \text{using}\quad u=y-2x+3\)

29. \(\dfrac{dy}{dx}= \dfrac{1-x-y}{x+y}\quad \text{using}\quad u=x+y\)

30. \(\dfrac{dy}{dx}= \sin(x+y)\quad \text{using}\quad u=x+y\)

31. \(\dfrac{dy}{dx}= 1+e^{y-x+5}\quad \text{using}\quad u=y-x+5\)

In Exercises 32-33, solve the differential equation using the indicated substitution. 

32. \(\dfrac{dy}{dx}= \cos(x+y), \quad u(0)=\dfrac{\pi}{4}\quad \text{using}\quad u=x+y\)

33. \(\dfrac{dy}{dx}= \dfrac{3x+2y}{3x+2y+2}\quad (y(-1)=-1 \quad\text{using}\quad u=x+y\)

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