Question
Question: The solution of the differential equation \(xdy-ydx=\left( \sqrt{{{x}^{2}}+{{y}^{2}}} \right)dx\) is...
The solution of the differential equation xdy−ydx=(x2+y2)dx is
(A). y−x2+y2=Cx2
(B). y+x2+y2=Cx2
(C). y+x2+y2+Cx2=0
(D). None of these
Solution
Hint: Any differential equation would be of homogeneous type if the given function F (x, y) follows the equation F(λx,λy)=λnF(x,y) where n is an integer. And we can solve any differential equation in variables ‘x’ and ‘y’ by putting y = vx or x = vy where ‘v’ is another variable. Now, try to solve it. And use the integration of x2+a21given as ∫x2+a2dx=log(x+x2+a2).
Complete step-by-step solution -
Given differential equation is,
xdy−ydx=(x2+y2)dx.............(i)
On dividing the whole equation (i) by ‘dx’ we get,
xdxdy−y=(x2+y2)ordxdy=xx2+y2+y...........(ii)
Now, we need to observe the type of differential equation i.e. variable separable or linear or homogeneous equation.
Here, we can observe that we cannot separate the variables ‘x’ and ‘y’ and cannot form a linear relation as well.
So, let us check whether the differential equation is homogeneous or not?
Let F(x,y)=xy+x2+y2
Replace (x,y) by (λx,λy) and try to get an equation of type λnF(x,y).
Hence,
F(λx,λy)=λxλy+(λx)2+(λy)2F(λx,λy)=λxλy+λx2+y2F(λx,λy)=λ∘F(x,y)
Hence, the given differential equation is of homogeneous type.
So, take y = vx in equation (ii).
Now, differentiating y = vx, we get;
dxdy=vdxdx+xdxdv
Where, we used relation
dxd(u.v)=udxdu+vdxdu
Hence, we get
dxdy=v+xdxdv..............(iii)
So, equation (ii) can be written as
v+xdxdv=xx2+(vx)2+vxv+xdxdv=xx1+v2+vxv+xdxdv=1+v2+vxdxdv=1+v2..............(iv)
Now, it’s a simple variable separable differential equation and can be written as,
∫1+v2dv=∫x1dx...........(v)
Now, we know,
∫a2+x21dx=loge(x+a2+x2)∫x1dx=logex
Hence, equation (v) can be written as,
loge(1+1+v2)=logex+logec
Where logec is a constant.
Now, we can use property of logarithm as,
logam+logan=loga(mn)
So, we get
loge(1+1+v2)=logexc
And hence,
1+1+v2=cx
Now, putting the value of v=xy by relation y = vx. Hence, above equation can be written as’
1+1+(xy)2=cx1+xx2+y2=cxorx+x2+y2=cx2
And hence, option (B) is the correct answer.
Note: Calculation is the important side of the question.
Observation is the key of the differential equations. One can use x = vy relation as well for solving the given equation.
One may get confused why constant term is taken in log in solution.
Reason is simple, for the simplicity of the equation so that we can remove ‘log’ from the solution part. And we can use logec or C or tan−1c etc. for writing the constant part in indefinite integral.