Solve the following differential equation by using...

Transcript

00:01 In this problem we are going to solve the following differential equation by using integration factor.

00:10 So let us remember the general rule for the method of integration factors.

00:18 If you are given such an equation y prime plus p of x some function of x times y equal to some other function of x, g of x.

00:28 Then the solution is this integration factor mu times t of x d x plus some constant divide by the integration factor again here mu of x is calculated by exponential integral px d x so the given equation x plus 6 y prime equal to 9x plus y so let us try to write this in this form you so that we know what p and g are if we divide both sides by x plus 6 we obtain y prime minus 1 over x plus 6 times y equal to 9 x over x plus 6 so it is clear that this guy is p this is g now mu is given by exponential integral p d x so we have exponential integral minus 1 over x plus 6 d x and this equal to exponential minus ln x plus 6 we don't need the absolute value sign here because we assume that x is greater than minus 6 as indicates in the problem statement so if we simplify this we obtain 1 over x plus 6 now let us compute this integral mu times g d x we have 1 over x plus x plus 6 9x 9x over x plus 6 d x this equal to 9 times x d x x squared plus 12 x plus 36 i'm going to divide the numerator uh multiple the numerator and the denominator by 2 so i have this form that's turn 6 and then i will add 12 to the and also subtract it from the numerator.

03:15 And i will separate the two integrals.

03:18 So i have 2x plus 12 over x squared plus 12x plus 36 dx minus 12 dx plus 6 squared.

03:36 Now these are really elementary integrals...

Question

Solve the following differential equation by using integrating factors. $x^2y' = xy - 8\ln x$, $y(1) = 24$ $4\ln(x) + 2 + 22x$

Question

Solve the following differential equation by using integrating factors. $x^2y' = xy - 8\ln x$, $y(1) = 24$ $4\ln(x) + 2 + 22x$

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