JEE Mains Maths · Differential Equations

Forming a Differential Equation

Going from a family of curves, or a given solution, to the differential equation it satisfies: eliminate the arbitrary constants by differentiating, then read off the order and degree.

All Differential Equations notes JEE Mains Maths strategy

Why this matters

Eight PYQs, seven of them multiple choice. Half give a family of circles or parabolas and ask for its equation; the rest give one solution or a rule for f and ask which equation it satisfies, or for the order and degree. Two ideas cover the page.

Concept 1 of 2: Eliminating the constants of a family

A family with n arbitrary constants satisfies a differential equation of order n. Differentiate the family n times, then use the equations to remove every constant. For circles through two fixed points, or parabolas with a fixed axis direction, write the general member with as few constants as the conditions allow before differentiating.

Definition

Count the free constants: that is the order of the equation.
Differentiate as many times as there are constants.
Solve for the constants from the derivatives and substitute back.
Circles through the origin with centre on $y=x$ : $x^2+y^2+gx+gy=0$ , one constant.

Order = number of constants

y=f(x;c_1,\dots,c_n)\ \Rightarrow\ F\big(x,y,y',\dots,y^{(n)}\big)=0

Worked example

Find the differential equation of the family

y=cx^2

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 28 July 2022 · Q162Moderate

Example 1 · Differential Equations · Forming a Differential Equation

The differential equation of the family of circles passing through the points

(0,2)

and

(0, - 2)

Use the conditions first

A general circle has three constants. Conditions such as 'through the origin' or 'centre on

y=x

' remove some of them before you differentiate; skipping that step gives an equation of too high an order.

Drill 3 more on eliminating the constants of a family

Concept 2 of 2: Order, degree, and the equation a solution satisfies

The order is the highest derivative present. The degree is the power of that highest derivative once the equation is free of radicals and fractions in the derivatives, so clear any square root first. When a question gives a solution, or a rule the function obeys, differentiate it until the equation in the options appears.

Definition

Order: the highest derivative.
Degree: the power of the highest derivative after removing radicals in the derivatives.
$y=A\cos(k\ln x)+B\sin(k\ln x)$ satisfies $x^2y''+xy'+k^2y=0$ .
$f(xy)=f(x)f(y)$ with $f'(1)=k$ gives $xf'(x)=kf(x)$ .

Degree after clearing radicals

y=x y'+\sqrt{1+y'^2}\ \Rightarrow\ (y-xy')^2=1+y'^2\ :\ \text{order }1,\ \text{degree }2

Worked example

Find the order and degree of

\left(\frac{d^2y}{dx^2}\right)^{3}+\left(\frac{dy}{dx}\right)^{4}=x

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 27 June 2022 · Q68Moderate

Example 2 · Differential Equations · Forming a Differential Equation

\cos^{- 1}\left( \frac{y}{2} \right)=\log_{e}\left( \frac{x}{5} \right)^{5},|y| < 2

, then :

Clear the radical before counting

y=xy'+\sqrt{1+y'^2}

looks like degree 1, but squaring to remove the root gives

y'^2

terms, so the degree is 2. The degree is read only after the radical is gone.

Drill 3 more on order, degree, and the equation a solution satisfies

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Eliminating the constants of a family
Order = number of constants
$y=f(x;c_1,\dots,c_n)\ \Rightarrow\ F\big(x,y,y',\dots,y^{(n)}\big)=0$
Order, degree, and the equation a solution satisfies
Degree after clearing radicals
$y=x y'+\sqrt{1+y'^2}\ \Rightarrow\ (y-xy')^2=1+y'^2\ :\ \text{order }1,\ \text{degree }2$

Watch out for (2)

Use the conditions first
→ Eliminating the constants of a family
Clear the radical before counting
→ Order, degree, and the equation a solution satisfies

Test yourself on Differential Equations

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Differential Equations chapter test Drill all 8 questions