Answer to Question #308126 in Differential Equations for Papi Chulo

Question #308126

Solve the separable differential equation for.



dy/dx= [1+x] divided by [xy^15]






Use the following initial condition: y(1)=5


y^16=?



.


1
Expert's answer
2022-03-10T18:18:16-0500

Let us solve the separable differential equation

dydx=1+xxy15\frac{dy}{dx}=\frac {1+x}{xy^{15}}

which is equivalent to

y15dy=1+xxdxy^{15}dy=\frac {1+x}{x}dx

It follows that

y15dy=1+xxdx=(1x+1)dx.\int y^{15}dy=\int\frac {1+x}{x}dx=\int(\frac{1}x+1)dx.

Therefore, y1616=lnx+x+C.\frac{y^{16}}{16}=\ln|x|+x+C.

Since y(1)=5,y(1)=5, we get that 51616=1+C,\frac{5^{16}}{16}=1+C, ànd hence C=516161.C=\frac{5^{16}}{16}-1.

We conclude that the solution is

y1616=lnx+x+516161\frac{y^{16}}{16}=\ln|x|+x+\frac{5^{16}}{16}-1 or

y16=16lnx+16x+51616y^{16}=16\ln|x|+16x+5^{16}-16



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