Answer:
The general solution of the differential equation is
The equation of the solution through the point (x,y)=(7,1) is
The equation of the solution through the point (x,y)=(0,-3) is
Step-by-step explanation:
This differential equation is a separable first-order differential equation.
We know this because a first order differential equation (ODE) is called a separable equation if the function
can be factored into the product of two functions of <em>x</em> and <em>y</em>
where<em> p(x) </em>and<em> h(y) </em>are continuous functions. And this ODE is equal to
To solve this differential equation we rewrite in this form:
And next we integrate both sides
So
We can subtract constants
An explicit solution is any solution that is given in the form . That means that the only place that y actually shows up is once on the left side and only raised to the first power.
An implicit solution is any solution of the form which means that y and x are mixed (<em>y</em> is not expressed in terms of <em>x</em> only).
The general solution of this differential equation is:
- To find the equation of the solution through the point (x,y)=(7,1)
We find the value of the with the help of the point (x,y)=(7,1)
Plug this into the general solution and then solve to get an explicit solution.
We need to check the solutions by applying the initial conditions
With the first solution we get:
With the second solution we get:
Therefore the equation of the solution through the point (x,y)=(7,1) is
- To find the equation of the solution through the point (x,y)=(0,-3)
We find the value of the with the help of the point (x,y)=(0,-3)
Plug this into the general solution and then solve to get an explicit solution.
We need to check the solutions by applying the initial conditions
With the first solution we get:
With the second solution we get:
Therefore the equation of the solution through the point (x,y)=(0,-3) is