Problem statement: Find the solution to the differential equation.
1) dy/dx= x^3-2y/x solution is: y= c/x^2 + x^3/5 the equation is linear but i do not know the steps.
The same goes for the following:
2. (x+y)dx - (x-y)dy=0 Solution is: arctan(y/x) - ln (square root of)x^2+y^2=c
6. xdy/dx +xy=1-y y(1)=0 solution is: y= x^-1(1-e^1-x)
8. xdy/dx +2y= sinx/x y(2)=1 solution is: (4+cos2-cosx)/x^2
12. dy/dx +y = 1/1+e^x solution is: y= ce^-x + e^-xln(1+e^x)
15. (e^x +1)dy/dx= y-ye^x solution is: y= c/cosh^2(x/2)
17. dy/dx= e^2x + 3y solution is: y= ce^3x-e^2x
20. y`= e^x+y solution is: e^x + e^-y=c
22. dy/dx= x^2-1/y^2+1 y(-1)=1 solution is: y^3+3y-x^3+3x=2
30. dy/dx= y^3/1-2xy^2 y(0)=1 solution is: xy^2 -ln[y]=0
31. (x^2y+xy-y)dx + (x^2y-2x^2)dy=0 solution is:[x+ln[x]+x^-1+y-2ln[y]=c
Please show me how you solve these problems. I know that 15,20,22,&31 are separable. The rest are linear except for 2, its homogeneous.
The Attempt at a Solution
For linear equations I am to find the integrating factor which is e to the power of the coefficient of y mulitplied by t. (e^xt) Then I multiply this factor with the equation and integrate to find the value of y. For seperable I am supposed to split the equation with x on one side and y on the other then integrate to find y. I tried doing that but could not figure out how to get the solutions that my professors gave me. Any help please?