Show that the differential equation $$\frac{dy}{dx} =\frac{e^x+x}{\sin y+2}$$ has a solution satisfying $y(0) = \pi$. To do this, separate variables and integrate to get an equation implicitly relating $x$ and $y$.
Asked
Active
Viewed 194 times
2 Answers
3
$$ \frac {dy}{dx} = \frac {e^x + x}{\sin y + 2} \\ (\sin y + 2) dy = (e^x + x) dx \\ \int (\sin y + 2) dy = \int (e^x + x) dx \\ -\cos y + 2y = e^x + \frac {x^2}2 + C $$ Now, substitute $y(0) = \pi$ $$ 1 + 2\pi = 1 + C $$ from which you can find that $C = 2\pi$.
Final answer is $$ 2y - \cos y = e^x + \frac {x^2}2 + 2\pi $$ is a solution of a given ODE and satisfies $y(0) = \pi$.
Kaster
- 9,722
-
Why is there a C? Isn't after integral both side, and there are two C at the left and right, then the C will be cancelled out? And how you get e^x+x^2/2 = 1? – Karen Jun 11 '13 at 08:34
-
@Karen You get $C_1$ on one side and $C_2$ on the other and the difference is $C$ - the two constants cannot be assumed to be the same. – Mark Bennet Jun 11 '13 at 08:40
-
oh right! my mistake! – Karen Jun 11 '13 at 08:42
-
then where did your other C go? – Karen Jun 11 '13 at 08:42
-
@Karen $C$ was determined based on your initial condition $y(0) = \pi$. It's all written in the post. – Kaster Jun 11 '13 at 09:03
0
Here is a start. You need to use the existence and uniqueness theorem for first order ode.
$$ \frac{dy}{dx} =\frac{e^x+x}{\sin y+2}=F(x,y).$$
Now, just find $F_y=\frac{\partial F}{\partial y}$ and apply the theorem.
Mhenni Benghorbal
- 47,431