Question

Find the solution of the differential equation that satisfies the given initial condition.
(Check my work please :D ).

Answer 1

\[y'tanx = 9a+y, y(\pi/3)=9a, o < x < \pi/2\]a is a constant.

Answer 2

Multiply both sides by dx/ (tanx)(a+y)
\[\frac{ dy }{ a+y }=\frac{ 9dx }{ tanx }\]\[\int\limits_{}^{}\frac{ 1 }{ a+y }dy=9\int\limits_{}^{}cotxdx\]\[\ln \left| a+y \right|=9\ln \left| sinx \right|+K\]

Replace K with lnC with positive constant
\[\ln \left| a+y \right|=9\ln(sinx)+lnC\]\[\ln \left| a+y \right|=9\ln(Csinx)\]

Answer 3

Raise power of e to both sides,
\[\left| a+y \right|=9Csinx\]
When x=pi/3, y = 9a
\[\left| a+9a \right|= 9C \frac{ \sqrt{3} }{ 2 }\]\[\frac{ 2 }{ \sqrt{3} }(\frac{ 1 }{ 9})(10\left| a \right|)= C\]\[\frac{ 20 }{ 9\sqrt{3} }\left| a \right|=C\]\[\left| a+y \right|=9(\frac{ 20 }{ 9\sqrt{3} }\left| a \right|)sinx\]\[y = \frac{ 20 }{ \sqrt{3} }asinx-a\]

Answer 4

Got this answer wrong, what do?

Answer 5

9a+y is not same as 9(a+y)

Answer 6

yup just a simple factor problem.

Answer 7

\[\ln \left| a+y \right|=9\ln \left| sinx \right|+C\]\[\left| a+y \right|=e^{9\ln \left| sinx \right|+C} = e^{9\ln \left| sinx \right|}*e^C\]\[\left| a+y \right|=9\left| sinx \right|e^C, K = \pm e^C\]\[a+y = 9sinx K => y = 9sinxK-a\]\[x = \pi/3, y = 9a\]\[a + 9a = K \sin \frac{ \pi }{ 3 }, 10a = \frac{\sqrt{3}}{ 2 }K, K = \frac{ 20a }{ \sqrt{3} }\]
Thus, \[y = \frac{ 180a }{ \sqrt{3} }sinx-a\]

Answer is still wrong, at what part did I mess up on with 9a+y not being the same as 9(a+y)?

Answer 8

@Zenmo The very first part, sadly:

y′tanx=9a+y
dytanx = (9a+y)dx

dy/(9a+y) = cot(x) dx
In(9a+y) = In|sin(x)| + K = ln(sin(x)) + K for some constant C since sin is positive over (0,pi/2)

***Note carefully we never factored the 9 out to start with! The nine belongs to the constant a, not the variable y so there is no need to factor is out!)


9a+y = exp(ln(sinx)+K)
9a + y = sin(x).exp(K)
y = exp(K). sin(x) - 9a = Csin(x) - 9a for some positive constant C

Sub in x = pi/3, y = 9a
9a = Csin(pi/3) - 9a
C = 18a/(sqrt(3)/2) = 36a/sqrt(3)

So y = 36a/(sqrt(3) * sin(x) - 9a = 9a(4sin(x)/(sqrt(3)) - 1)