Solve The Differential Equation Xy 2y X 2

x y 2y = x 2.

Divide x on both sides.

y2y / x = x (1)

First order linear equation of form y + p (x) y = q (x)

p (x) = 2 / x.

q (x) = x

The integration factor is e ¢ p "p (x) dx = e 2" 2 / x dx = e (2ln x) = e (ln x (2)) = 1 / x 2

Multiply the equation (1) by 1 / x 2.

y (1 / x 2) 2y (1 / x 3) = 1 / x (2)

To the left of (1) is d / dx (y times integration factor) or (y / x 2)

(y / x 2) = 1 / x.

Merge both sides

("(y / x 2) dx =" 1 / x dx

y / x 2 = lnx + C.

y = x 2 ln x + C x 2.

the answer:

y (2y / x) = x.

We know that the solution of all linear equations is y = y_h + y_p. I have

The solution of y_h is to find the right side of z. To explain

y = 2y / x y / y = 2 / x ln (y) = 2 * ln | x | + K.

y = (and K) * (x 2) set and K = C.

So now we know y_h = C * (x 2)

Now solve for y_p creation action C, that is.

(d / dx C (x) * (x 2)) 2 * (C (x) * (x 2) / x) = x

C (x) * (x 2) + 2x * C (x) 2 * C (x) * x = x

C (x) * (x 2) = x C (x) * (x) = 1 C (x) = 1 / x C (x) = ln | x |

The final answer is:

y = C * (x -2) + (x -2) * ln | x |

y = (x 2) * (C + ln | x |)

Solve The Differential Equation Xy 2y X 2

Solve The Differential Equation Xy 2y X 2

x y 2y = x 2

Divide both sides by x.

y2y / x = x (1)

The first sequential linear equation of the form y + p (x) y = q (x)

p (x) = 2 / x

q (x) = x

The integration factor is e ˆ "p (x) dx = e ˆÂ" 2 / x dx = e (2ln x) = e (ln x (2)) = 1 / x 2

Multiply the equation (1) by 1 / x 2.

y (1 / x 2) 2y (1 / x 3) = 1 / x (2)

To the left of (1) d / dx (y fold integration element) or (y / x 2)

(y / x 2) = 1 / x

Merge both sides

ˆ "(y / x 2) dx = ˆÂ" 1 / x dx

y / x 2 = lnx + C

y = x 2 ln x + C x 2

y (2y / x) = x

We know that the solution of all linear equations is y = y_h + y_p. I have

The solution of y_h is to find the right side of z. To explain

y = 2y / x y / y = 2 / x ln (y) = 2 * ln | x | + K

y = (and K) * (x 2) set and K = C

So now we know y_h = C * (x 2)

Now end the y_p C-Action, then

(d / dx C (x) * (x 2)) 2 * (C (x) * (x 2) / x) = x

C (x) * (x 2) + 2x * C (x) 2 * C (x) * x = x

C (x) * (x 2) = x C (x) * (x) = 1 C (x) = 1 / x C (x) = ln | x |

The final answer is:

y = C * (x 2) + (x 2) * ln | x |

y = (x 2) * (C + ln | x |)

Solve The Differential Equation Xy 2y X 2