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:The general solution of the differential equation is [tex]\frac{49y^{2} }{2}-\frac{x^{2} }{2} = c_{3}[/tex]The equation of the solution through the point (x,y)=(7,1) is [tex]y=\frac{x}{7}[/tex]The equation of the solution through the point (x,y)=(0,-3) is [tex]\:y=-\frac{\sqrt{441+x^2}}{7}[/tex]Step-by-step explanation:This differential equation [tex]\frac{dy}{dx}=\frac{x}{49y}[/tex] is a separable first-order differential equation.We know this because a first order differential equation (ODE) [tex]y' =f(x,y)[/tex] is called a separable equation if the function [tex]f(x,y)[/tex] can be factored into the product of two functions of x and y[tex]f(x,y)=p(x)\cdot h(y)[/tex] where p(x) and h(y) are continuous functions. And this ODE is equal to [tex]\frac{dy}{dx}=x\cdot \frac{1}{49y}[/tex]To solve this differential equation we rewrite in this form:[tex]49y\cdot dy=x \cdot dx[/tex]And next we integrate both sides[tex]\int\limits {49y} \, dy=\int\limits {x} \, dx[/tex][tex]\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}[/tex][tex]\int\limits {x} \, dx=\frac{x^{2} }{2} +c_{2}[/tex]So[tex]\int\limits {49y} \, dy=\int\limits {x} \, dx\\\frac{49y^{2} }{2} + c_{1} =\frac{x^{2} }{2} +c_{2}[/tex]We can subtract constants [tex]c_{3}=c_{2}-c_{1}[/tex][tex]\frac{49y^{2} }{2} =\frac{x^{2} }{2} +c_{3}[/tex]An explicit solution is any solution that is given in the form [tex]y=y(t)[/tex]. 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  [tex]f(x,y)=g(x,y) [/tex] 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:[tex]\frac{49y^{2} }{2}-\frac{x^{2} }{2} = c_{3}[/tex]To find the equation of the solution through the point (x,y)=(7,1)We find the value of the [tex]c_{3}[/tex] with the help of the point (x,y)=(7,1)[tex]\frac{49*1^2\:}{2}-\frac{7^2\:}{2}\:=\:c_3\\c_3 = 0[/tex]Plug this into the general solution and then solve to get an explicit solution.[tex]\frac{49y^2\:}{2}-\frac{x^2\:}{2}\:=\:0[/tex][tex]\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)}[/tex][tex]y=\frac{x}{7},\:y=-\frac{x}{7}[/tex]We need to check the solutions by applying the initial conditionsWith the first solution we get:[tex]y=\frac{x}{7}=\\1=\frac{7}{7}\\1=1\\[/tex]With the second solution we get:[tex]\:y=-\frac{x}{7}\\1=-\frac{7}{7}\\1\neq -1[/tex]Therefore the equation of the solution through the point (x,y)=(7,1) is [tex]y=\frac{x}{7}[/tex]To find the equation of the solution through the point (x,y)=(0,-3)We find the value of the [tex]c_{3}[/tex] with the help of the point (x,y)=(0,-3)[tex]\frac{49*-3^2\:}{2}-\frac{0^2\:}{2}\:=\:c_3\\c_3 = \frac{441}{2}[/tex]Plug this into the general solution and then solve to get an explicit solution.[tex]\frac{49y^2\:}{2}-\frac{x^2\:}{2}\:=\:\frac{441}{2}[/tex][tex]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}[/tex]We need to check the solutions by applying the initial conditionsWith the first solution we get:[tex]y=\frac{\sqrt{441+x^2}}{7}\\-3=\frac{\sqrt{441+0^2}}{7}\\-3\neq 3[/tex]With the second solution we get:[tex]y=-\frac{\sqrt{441+x^2}}{7}\\-3=-\frac{\sqrt{441+0^2}}{7}\\-3=-3[/tex]Therefore the equation of the solution through the point (x,y)=(0,-3) is [tex]\:y=-\frac{\sqrt{441+x^2}}{7}[/tex]