Test: Differential Equations: Order & Degree(20 Oct) - JEE MCQ

To calculate the degree or the order of a differential equation, the powers of derivatives should be an integer.
On squaring both sides, we get a differential equation with the integral power of derivatives.

⇒ Order (the highest derivative) = 2
⇒ Degree (the power of highest degree) = 2

y = Acos(αx) + Bsin(αx)

dy/dx = -Aαsin(αx) + Bαcos(αx)

d²y/dx² = -Aα²cos(αx) - Bα²sin(αx)

= -α²(Acos(αx) + Bsin(αx))

= -α² * y

d²y/dx² + α²*y = 0

y = Ae^2x + Be^-2x 
dy/dx = 2Ae^2x – 2Be^-2x
d²y/dx² = 4Ae^2x + 4Be^-2x
= 4*y
d²y/dx² – 4y = 0

Given equation is : 

(dy/dx)² + 1/(dy/dx) = 1

((dy/dx)³ + 1)/(dy/dx) = 1

(dy/dx)³ +1 = dy/dx

So, final equation is

(dy/dx)³ - dy/dx + 1 = 0

So, degree = 3

The answer is C. We eliminate constants.

We have

y=ax+x²

Differentiating with respect to x,

Order of the D.E. is 3
Order of a differential equation is the order of the highest derivative present in the equation.

Equation of ellipse :

 x²/a² + y²/b² = 1

Differentiation by x,

2x/a² + (dy/dx)*(2y/b²) = 0

dy/dx = -(b²/a²)(x/y)

-(b²/a^2) = (dy/dx)*(y/x) ----- eqn 1

Again differentiating by x,

d²y/dx² = -(b²/a²)*((y-x(dy/dx))/y²)

Substituting value of -b²/a² from eqn 1

d²y/dx² = (dy/dx)*(y/x)*((y-x(dy/dx))/y²)

d²y/dx² = (dy/dx)*((y-x*(dy/dx))/xy)

(xy)*(d²y/dx²) = y*(dy/dx) - x*(dy/dx)²

(xy)*(d²y/dx²) + x*(dy/dx)²- y*(dy/dx) = 0

Solving second order differential equation with variable coefficients becomes a bit lengthy and complicated. So, its better to check by options.

On checking option A :

y = A/x + B

dy/dx = -A/x²

d²y/dx² = (2A)/x³

So,

 d²y/dx² + (2/x)*(dy/dx) = 0

(2A)/x³ + (2/x)*((-A)/x²) = 0

(2A - 2A)/x³ = 0

0 = 0

LHS = RHS

3*(d²y/dx²) = [1+(dy/dx)²]³/²
On squaring both side,
9*(d²y/dx²)² = [1+(dy/dx)²]³
The order of the equation is 2. The power of the term determining the order determines the degree.
So, the degree is also 2.

The highest order derivative here is y’’. Therefore the order of the differential equation=2.
The highest power of the highest order derivative here is 3. Therefore the order of the differential equation=3.