Ordinary Differential Equations/Separable 4

Existence problemsEdit

1) f(x,y) has no discontinuities, so a solution exists. \frac{\partial {f} }{\partial {y} } has no discontinuities, so the solution is unique.

2) f(x,y) is not defined for the point (-1,10) because ln(x) is not defined. So no solution exists.

3) f(x,y) has discontinuities at y=1 and -1, but not at 0 so a solution exists. \frac{\partial {f} }{\partial {y} } has no discontinuities at (0,16) so the solution is unique.

4) f(x,y) has discontinuities at y<0, but not at 1 so a solution exists. \frac{\partial {f} }{\partial {y} } is discontinuous at 1, so the solution is not unique

5) f(x,y) has discontinuities at -3 and -4, but not at 0 so a solution exists. \frac{\partial {f} }{\partial {y} } has no discontinuities at (5,9) so the solution is unique.

6) f(x,y) has a discontinuity at x=5, so no solution exists.

Separable equationsEdit

7) y'=y^3sec^2(x)

\frac{dy}{y^3}=sec^2(x)dx

\int \frac{dy}{y^3}=\int sec^2(x)dx

-\frac{1}{2y^2}=tan(x)+C

y=-\frac{1}{\sqrt{(2tan(x)+C)}}


8) y'=\frac{5y^2+6}{y}

\frac{ydy}{5y^2+6}=dx

\int \frac{ydy}{5y^2+6}=\int dx

\frac{1}{10}ln(5y^2+6)=x+C

y=\pm\sqrt{Ce^{10x}-\frac{6}{5}}


9) y'=x^3/y^3

y^3dy=x^3dx

\int y^3dy=\int x^3dx

\frac{1}{4}y^4=\frac{1}{4}x^4+C

y=(x^4+C)^{\frac{1}{4}}


10) y'=x^2+3x-9

dy=(x^2+3x-9)dx

\int dy=\int (x^2+3x-9)dx

y=\frac{1}{3}x^3+\frac{3}{2}x^2-9x+C


11) y'=cos(y)/sin(y)

\frac{sin(y)dy}{cos(y)}=dx

\int \frac{sin(y)dy}{cos(y)}=\int dx

-ln(cos(y))=x+C

y=arccos(Ce^x)


12) y'=\frac{cos(x)}{sin(y)}

sin(y)dy=cos(x)dx

\int sin(y)dy=\int cos(x)dx

-cos(y)=sin(x)+C

y=arccos(-sin(x)+C)

Initial value problemsEdit

13) y'=cos(x)+sin(x),y(0)=1

dy=(cos(x)+sin(x))dx

\int dy=\int (cos(x)+sin(x))dx

y=sin(x)-cos(x)+C

1=sin(0)-cos(0)+C=0-1+C=C-1

C=2

y=sin(x)-cos(x)+2


14) y'=7y^2,y(5)=9

\frac{dy}{y^2}=7dx

\int \frac{dy}{y^2}=\int 7dx

-\frac{1}{y}=7x+C

y=\frac{1}{-7x+C}

9=\frac{1}{-7*5+C}

C=\frac{316}{9}

y=\frac{1}{-7x+\frac{316}{9}}

Last modified on 27 November 2010, at 05:41