Question

Solve the differential equation dy/dx=x/49y. Find an implicit solution and put your answer in the following form: = constant. help (formulas) Find the equation of the solution through the point (x,y)=(7,1). help (equations) Find the equation of the solution through the point (x,y)=(0,−3). Your answer should be of the form y=f(x). help (equations)

Answer 1

Answer:

The general solution of the differential equation is $\frac{49y^{2} }{2}-\frac{x^{2} }{2} = c_{3}$

The equation of the solution through the point (x,y)=(7,1) is $y=\frac{x}{7}$

The equation of the solution through the point (x,y)=(0,-3) is $\:y=-\frac{\sqrt{441+x^2}}{7}$

Step-by-step explanation:

This differential equation $\frac{dy}{dx}=\frac{x}{49y}$ is a separable first-order differential equation.

We know this because a first order differential equation (ODE) $y' =f(x,y)$ is called a separable equation if the function $f(x,y)$ can be factored into the product of two functions of x and y

$f(x,y)=p(x)\cdot h(y)$ where p(x) and h(y) are continuous functions. And this ODE is equal to $\frac{dy}{dx}=x\cdot \frac{1}{49y}$

To solve this differential equation we rewrite in this form:

$49y\cdot dy=x \cdot dx$

And next we integrate both sides

$\int\limits {49y} \, dy=\int\limits {x} \, dx$

$\mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=\frac{x^{a+1}}{a+1}\\\int\limits {49y} \, dy=\frac{49y^{2} }{2} + c_{1}$

$\int\limits {x} \, dx=\frac{x^{2} }{2} +c_{2}$

So

$\int\limits {49y} \, dy=\int\limits {x} \, dx\\\frac{49y^{2} }{2} + c_{1} =\frac{x^{2} }{2} +c_{2}$

We can subtract constants $c_{3}=c_{2}-c_{1}$

$\frac{49y^{2} }{2} =\frac{x^{2} }{2} +c_{3}$

An explicit solution is any solution that is given in the form $y=y(t)$. 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  $f(x,y)=g(x,y)$ which means that y and x are mixed (y is not expressed in terms of x only).

The general solution of this differential equation is:

$\frac{49y^{2} }{2}-\frac{x^{2} }{2} = c_{3}$

• To find the equation of the solution through the point (x,y)=(7,1)

We find the value of the $c_{3}$ with the help of the point (x,y)=(7,1)

$\frac{49*1^2\:}{2}-\frac{7^2\:}{2}\:=\:c_3\\c_3 = 0$

Plug this into the general solution and then solve to get an explicit solution.

$\frac{49y^2\:}{2}-\frac{x^2\:}{2}\:=\:0$

$\mathrm{Add\:}\frac{x^2}{2}\mathrm{\:to\:both\:sides}\\\frac{49y^2}{2}-\frac{x^2}{2}+\frac{x^2}{2}=0+\frac{x^2}{2}\\Simplify\\\frac{49y^2}{2}=\frac{x^2}{2}\\\mathrm{Multiply\:both\:sides\:by\:}2\\\frac{2\cdot \:49y^2}{2}=\frac{2x^2}{2}\\Simplify\\9y^2=x^2\\\mathrm{Divide\:both\:sides\:by\:}49\\\frac{49y^2}{49}=\frac{x^2}{49}\\Simplify\\y^2=\frac{x^2}{49}\\\mathrm{For\:}x^2=f\left(a\right)\mathrm{\:the\:solutions\:are\:}x=\sqrt{f\left(a\right)},\:\:-\sqrt{f\left(a\right)}$

$y=\frac{x}{7},\:y=-\frac{x}{7}$

We need to check the solutions by applying the initial conditions

With the first solution we get:

$y=\frac{x}{7}=\\1=\frac{7}{7}\\1=1$

With the second solution we get:

$\:y=-\frac{x}{7}\\1=-\frac{7}{7}\\1\neq -1$

Therefore the equation of the solution through the point (x,y)=(7,1) is $y=\frac{x}{7}$

• To find the equation of the solution through the point (x,y)=(0,-3)

We find the value of the $c_{3}$ with the help of the point (x,y)=(0,-3)

$\frac{49*-3^2\:}{2}-\frac{0^2\:}{2}\:=\:c_3\\c_3 = \frac{441}{2}$

Plug this into the general solution and then solve to get an explicit solution.

$\frac{49y^2\:}{2}-\frac{x^2\:}{2}\:=\:\frac{441}{2}$

$y^2=\frac{441+x^2}{49}\\\mathrm{For\:}x^2=f\left(a\right)\mathrm{\:the\:solutions\:are\:}x=\sqrt{f\left(a\right)},\:\:-\sqrt{f\left(a\right)}\\y=\frac{\sqrt{441+x^2}}{7},\:y=-\frac{\sqrt{441+x^2}}{7}$

We need to check the solutions by applying the initial conditions

With the first solution we get:

$y=\frac{\sqrt{441+x^2}}{7}\\-3=\frac{\sqrt{441+0^2}}{7}\\-3\neq 3$

With the second solution we get:

$y=-\frac{\sqrt{441+x^2}}{7}\\-3=-\frac{\sqrt{441+0^2}}{7}\\-3=-3$

Therefore the equation of the solution through the point (x,y)=(0,-3) is $\:y=-\frac{\sqrt{441+x^2}}{7}$

Answer 2

The given differential equation is of first order (as it involves only the first rate of function) and is Ordinary Differential Equation (ODE).

The answers would be:

• The solution of given differential equation is: $\dfrac{49y^2}{2} = \dfrac{x^2}{2} + c$

• The equation of solution through  the point (x,y)=(7,1) is:  $y = \dfrac{x}{7}$

• The equation of the solution through the point (x,y)=(0,−3) is: $y = - \dfrac{\sqrt{x^2 +441}}{7}$

Given that:

• The given differential equation is:   $\dfrac{dy}{dx} = \dfrac{x}{49y}$

• Two given points where equation of solution is to be found are: (7,1), and (0,-3)

How to solve first order differential equations?

The given differential equation is of first order (as it involves only the first rate of function) and is Ordinary Differential Equation (ODE).

Writing it in separated forms to do integration:

$\dfrac{dy}{dx} = \dfrac{x}{49y}\\\\49y\: dy = x\: dx$

Integrating on both sides, we get:

$\int49y \: dy = \int x dx\\\\\dfrac{49y^2}{2} = \dfrac{x^2}{2} + c$

Thus, the solution of given differential equation is: $\dfrac{49y^2}{2} = \dfrac{x^2}{2} + c$

Evaluation at given points

Evaluating at point (7,1)

$\dfrac{49y^2}{2} = \dfrac{x^2}{2} + c\\\\\dfrac{49\times 1^2}{2} = \dfrac{7^2}{2} + c\\\\c =0$

Equation of the solution through  (x,y)=(7,1) is calculated as:

$\dfrac{49y^2}{2} = \dfrac{x^2}{2} + 0\\\\y^2 = \dfrac{x^2}{49}\\\\y = \pm \dfrac{x}{7}$

Evaluating the equations at (7,1) to get correct equation:

$y = \dfrac{x}{7}\\1 = \dfrac{7}{7}\\1 = 1$

and

$y = -\dfrac{x}{7}\\1 = -\dfrac{7}{7}\\1 = -1$ which is incorrect.

Thus,

Thus the equation of solution through  the point (x,y)=(7,1) is:  $y = \dfrac{x}{7}$

Evaluating at point (0,-3)

$\dfrac{49y^2}{2} = \dfrac{x^2}{2} + c\\\\\dfrac{49\times (-3)^2}{2} = \dfrac{(0)^2}{2} + c\\\\c = \dfrac{441}{2}$

Thus, equation of the solution through  (x,y) = (0, -3) can be found as:

$\dfrac{49y^2}{2} = \dfrac{x^2}{2} + \dfrac{441}{2}\\\\y^2 = \dfrac{x^2 +441 }{49}\\\\y = \pm \dfrac{\sqrt{x^2+ 441}}{7}$

Testing the equations at point (0, -3)

$y = \dfrac{\sqrt{x^2+ 441}}{7}\\-3 = \dfrac{\sqrt{(0)^2+ 441}}{7}\\\\-3 = \dfrac{\sqrt{441}}{7} = \dfrac{21}{7} = 3\\-3 = 3$ which is incorrect.

and

$-3 = -\dfrac{\sqrt{(0)^2+ 441}}{7}\\\\-3 = -\dfrac{\sqrt{441}}{7} = -\dfrac{21}{7} = -3\\-3 = -3$

Thus. the equation of the solution through the point (x,y)=(0,−3) is: $y = - \dfrac{\sqrt{x^2 + 441}}{7}$

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