Show that # y^2 = (4x)(ax) = 4ax4x^2 # is a solution to the DE? # 2xy dy/dx = y^2  4x^2#
(portions of this question have been edited or deleted!)
(portions of this question have been edited or deleted!)
As the quoted Differential Equation is malformed, let us work back from the quoted solution to form the correct DE:
Correction of the Question:
If a solution is:
is a solution, then implicit differentiation gives:
Therefore the question should (presumably) be to solve the DE
Solution:
We have:
As suggested, let us perform the substitution:
Substituting into the (corrected) DE [A] we have:
This is now a First Order separable DE, so we can "separate the variables" to get:
And now we can integrate, to get:
Applying the initial conditions:
So our particular solution is:
Restoring the substitution:
By signing up, you agree to our Terms of Service and Privacy Policy
To show that (y^2 = (4x)(ax) = 4ax  4x^2) is a solution to the differential equation (2xy \frac{dy}{dx} = y^2  4x^2):

Calculate ( \frac{dy}{dx} ) and ( \frac{d^2y}{dx^2} ). [ \frac{dy}{dx} = \frac{d}{dx}(4ax  4x^2) = 4a  8x ] [ \frac{d^2y}{dx^2} = \frac{d}{dx}(4a  8x) = 8 ]

Plug (y^2 = 4ax  4x^2) and its derivatives into the given differential equation. [ 2xy \frac{dy}{dx} = 2x(4ax  4x^2)(4a  8x) = 2x(16ax  8ax^2  32x + 8x^2) = 32ax^2  16ax^3  64x^2 + 16x^3 ] [ = 16x^3  16ax^3  64x^2 + 32ax^2 = 16ax^3 + 32ax^2  64x^2 + 16x^3 ]
[ y^2  4x^2 = (4ax  4x^2)^2  4x^2 = (16a^2x^2  32ax^3 + 16x^4)  4x^2 ] [ = 16a^2x^2  32ax^3 + 16x^4  4x^2 ] [ = 16x^4  32ax^3 + 16a^2x^2  4x^2 ]
 Check if (2xy \frac{dy}{dx} = y^2  4x^2) is true for (y^2 = 4ax  4x^2). [ 16ax^3 + 32ax^2  64x^2 + 16x^3 = 16x^4  32ax^3 + 16a^2x^2  4x^2 ]
The equation holds true. Therefore, (y^2 = 4ax  4x^2) is a solution to the given differential equation.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
 What is the surface area produced by rotating #f(x)=sinxcosx, x in [0,pi/4]# around the xaxis?
 How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y =1/(x^2+1)#, x=0, x=1, y=0 revolved about the yaxis?
 How do you solve #y'+3y=0# given y(0)=4?
 What is the surface area of the solid created by revolving #f(x)=sqrt(x^3)# for #x in [1,2]# around the xaxis?
 What is the arc length of the curve given by #f(x)=1+cosx# in the interval #x in [0,2pi]#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7