Is it correct to say: Reading off the degree before removing radicals or fractional powers?

First make the equation a polynomial in the derivatives (square, cube, clear fractions), then read the power of the highest-order derivative.

Is it correct to say: Giving a degree when a derivative appears inside sin, cos, log or an exponential?

If the equation cannot be written as a polynomial in the derivatives, say 'degree is not defined'. The order is still defined.

Is it correct to say: Taking P from an equation whose dy/dx coefficient is not 1?

Divide through first. For x dy/dx + 2y = x², divide by x: P = 2/x, Q = x, so I.F. = x².

Is it correct to say: Writing y(I.F.) = ∫Q dx + c?

The integrating factor must be multiplied inside the integral too: y(I.F.) = ∫Q(I.F.) dx + c.

Is it correct to say: In a homogeneous equation, replacing dy/dx by dv/dx?

y = vx gives dy/dx = v + x dv/dx. Also remember to substitute v = y/x back at the end.

Is it correct to say: Applying Newton's law of cooling to θ instead of θ − θ₀?

The excess temperature θ − θ₀ decays exponentially, not θ itself. The body approaches room temperature, not 0°C.

📄 Formula sheet (PDF)

Differential Equations

Order and degree, formation of differential equations, variable separable, homogeneous and linear equations, and applications to growth, decay and Newton's law of cooling — Maharashtra HSC Maths Part II Ch 6

📐 15 formulas✏️ 4 examples🎯 6 practice🏫 Maharashtra State Board📚 Class 12✓ 2025–26 syllabus

💡

Board Exam Tips

→Classify the equation before solving: variables separable, reducible (a function of ax + by + c), homogeneous (numerator and denominator are homogeneous functions of the same degree) or linear (y and dy/dx appear only to the first power).
→For a linear equation, first make the coefficient of dy/dx equal to 1, then read off P and Q.
→Remember e^(log f(x)) = f(x) — most integrating factors simplify to something like x, 1/x, x² or sec x.
→Write the arbitrary constant once, immediately after integrating. When the other terms are logarithms, write it as log c — the final answer simplifies neatly.
→In application problems, first write the law as a differential equation, solve it, and only then use the given data to find the constants.
→For a particular solution, substitute the given condition into the general solution to find c, and write the final equation without c.

📐 Formulas(15)

Order and Degree

(\frac{d ^{2} y}{d x ^{2}})^{3} + (\frac{d y}{d x})^{4} + y = 0 \Rightarrow order 2, degree 3

Degree After Clearing Radicals

[1 + (\frac{d y}{d x})^{2}]^{3/2} = k \frac{d ^{2} y}{d x ^{2}} \Rightarrow [1 + (\frac{d y}{d x})^{2}]^{3} = k^{2} (\frac{d ^{2} y}{d x ^{2}})^{2}

Formation of a Differential Equation

n arbitrary constants \Rightarrow differentiate n times \Rightarrow DE of order n

General and Particular Solution

y = A cos x + B sin x is the general solution of \frac{d ^{2} y}{d x ^{2}} + y = 0

Variables Separable★ Board fav

\frac{d y}{d x} = \frac{f ( x )}{g ( y )} \Rightarrow \int g (y) d y = \int f (x) d x + c

Reducible to Variables Separable

\frac{d y}{d x} = f (a x + b y + c) : put a x + b y + c = v, \frac{d y}{d x} = \frac{1}{b} (\frac{d v}{d x} - a)

Homogeneous Function

f (λ x, λ y) = λ^{n} f (x, y)

Homogeneous Equation: Substitution★ Board fav

y = v x \Rightarrow \frac{d y}{d x} = v + x \frac{d v}{d x}

Linear Differential Equation

\frac{d y}{d x} + P y = Q

Symbol	Meaning
$P$	Coefficient of y, a function of x
$Q$	Right-hand side, a function of x

Integrating Factor

I.F. = e^{\int P d x}

Solution of a Linear Equation★ Board fav

y (I.F.) = \int Q (I.F.) d x + c

Linear Equation in x

\frac{d x}{d y} + P x = Q \Rightarrow x (I.F.) = \int Q (I.F.) d y + c, I.F. = e^{\int P d y}

Law of Natural Growth and Decay★ Board fav

\frac{d x}{d t} = k x \Rightarrow x = x_{0} e^{k t}

Symbol	Meaning
$x$	Amount present at time t
$x_{0}$	Initial amount (at t = 0)
$k$	Constant of proportionality (per unit time)

Half-Life

x_{0} e^{- k T} = \frac{x _{0}}{2} \Rightarrow T = \frac{l o g 2}{k}

Newton's Law of Cooling★ Board fav

\frac{d θ}{d t} = - k (θ - θ_{0}) \Rightarrow θ - θ_{0} = C e^{- k t}

Symbol	Meaning
$θ$	Temperature of the body at time t (°C)
$θ_{0}$	Temperature of the surroundings (°C), constant
$k$	Positive constant (per unit time)

✏️ Solved Examples

1Solved Exampleeasy3 steps

Find the order and degree of (i) (d³y/dx³)² + (d²y/dx²)³ + y = 0, (ii) y = x(dy/dx) + √(1 + (dy/dx)²).

(i) The highest derivative is d³y/dx³, and its power is 2. The equation is already a polynomial in the derivatives.

order 3, degree 2

2Solved Exampleboard4 steps

Solve dy/dx + y/x = x².

The equation is linear with P = 1/x and Q = x².

P = \frac{1}{x}, Q = x^{2}

3Solved Exampleboard5 steps

Solve (x² + y²) dx − 2xy dy = 0.

Rewrite as dy/dx. Numerator and denominator are both homogeneous of degree 2, so the equation is homogeneous.

\frac{d y}{d x} = \frac{x ^{2} + y ^{2}}{2 x y}

4Solved ExampleHOTS4 steps

A body at 80°C is placed in a room kept at 20°C. It cools to 50°C in 20 minutes. Using Newton's law of cooling, find its temperature after 40 minutes.

By Newton's law of cooling, θ − θ₀ = Ce^{−kt} with θ₀ = 20.

θ - 20 = C e^{- k t}

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Reading off the degree before removing radicals or fractional powers
✓First make the equation a polynomial in the derivatives (square, cube, clear fractions), then read the power of the highest-order derivative.
2
Giving a degree when a derivative appears inside sin, cos, log or an exponential
✓If the equation cannot be written as a polynomial in the derivatives, say 'degree is not defined'. The order is still defined.
3
Taking P from an equation whose dy/dx coefficient is not 1
✓Divide through first. For x dy/dx + 2y = x², divide by x: P = 2/x, Q = x, so I.F. = x².
4
Writing y(I.F.) = ∫Q dx + c
✓The integrating factor must be multiplied inside the integral too: y(I.F.) = ∫Q(I.F.) dx + c.
5
In a homogeneous equation, replacing dy/dx by dv/dx
✓y = vx gives dy/dx = v + x dv/dx. Also remember to substitute v = y/x back at the end.
6
Applying Newton's law of cooling to θ instead of θ − θ₀
✓The excess temperature θ − θ₀ decays exponentially, not θ itself. The body approaches room temperature, not 0°C.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Find the order and degree of d²y/dx² + sin(dy/dx) = 0.
Q2
Form the differential equation by eliminating A and B from y = Ae^{2x} + Be^{−2x}.
Q3
Solve dy/dx = e^{x + y}.
Q4
Solve dy/dx = (x + y)².
Q5
Solve dy/dx + y = e^{−x}.
Q6
A population grows at a rate proportional to its size and doubles in 25 years. In how many years will it become three times its original size?

📝 Notes

Differential Equations

A differential equation relates a function to its derivatives. This chapter asks you to describe one (order and degree), build one (formation), solve one (four standard types) and use one (growth, decay and cooling).

Order, degree and formation

The order is the order of the highest derivative. For the degree, first make the equation a polynomial in the derivatives — clear radicals and fractional powers — and then read the power of the highest-order derivative. To form an equation, differentiate the given family as many times as it has arbitrary constants and eliminate them; the result has order equal to the number of constants.

Identify the type first

Variables separable: can be written as $g(y)\,dy = f(x)\,dx$ . Integrate both sides.
Reducible: the right side is a function of $ax + by + c$ . Put $v = ax + by + c$ and the equation separates.
Homogeneous: $\frac{dy}{dx} = \frac{f(x, y)}{g(x, y)}$ with $f$ and $g$ homogeneous of the same degree. Put $y = vx$ .
Linear: $\frac{dy}{dx} + Py = Q$ . Multiply by $e^{\int P\,dx}$ . If the equation is linear in $x$ instead, use $\frac{dx}{dy} + Px = Q$ .

Some equations fit more than one type — choose whichever is quickest.

Applications

Growth and decay problems start from $\frac{dx}{dt} = kx$ , giving $x = x_0 e^{kt}$ . Newton's law of cooling starts from $\frac{d\theta}{dt} = -k(\theta - \theta_0)$ , giving $\theta - \theta_0 = Ce^{-kt}$ . In both, use the initial value to find the constant $C$ (or $x_0$ ), and a second piece of data to find $e^{-k}$ or $k$ . When the times are multiples of each other, powers such as $e^{-40k} = (e^{-20k})^2$ often avoid logarithms completely.