Differential Equations (Class 12 Maths): Types, Methods and Board-Style Solved Questions

Start with order and degree

Before any method, two definitions carry easy marks and are asked directly.

• Order is the order of the highest derivative present.
• Degree is the power of the highest-order derivative, after the equation has been made free of radicals and fractions in the derivatives.

That last clause is where students lose the mark. If a derivative sits under a square root or in a denominator, the equation must first be cleared — and if it cannot be expressed as a polynomial in the derivatives at all, the degree is not defined. Equations containing terms such as sin(dy/dx) or e^dy/dx fall into this category, and questions are set on exactly those cases.

Identify the type first — the decision that decides everything

Spend the first fifteen seconds of every question on this table. Students who begin manipulating immediately frequently start down the wrong route and lose more time recovering than the identification would have cost.

Variable separable

The most straightforward form. Collect all y terms with dy and all x terms with dx, then integrate.

Two habits matter here. Write the modulus inside the logarithm, and add the arbitrary constant immediately rather than at the end — the marks for it are routinely forgotten in the rush to finish.

Homogeneous equations

An equation is homogeneous when dy/dx can be expressed purely as a function of y/x. The substitution y = vx gives dy/dx = v + x(dv/dx), and after substitution the equation always becomes separable in v and x.

The step most often dropped is substituting back. After solving for v, you must return to y/x. An answer left in terms of v is incomplete.

Linear differential equations and the integrating factor

For an equation in the form dy/dx + Py = Q, with P and Q functions of x:

Three points cover almost all errors in this section:

1. Get the equation into standard form first. The coefficient of dy/dx must be 1. If it is not, divide through before identifying P and Q.
2. No constant when computing the integrating factor. Adding one there produces a constant multiplier that cancels anyway, but costs time and invites error.
3. If P and Q are functions of y instead of x, the equation is linear in x. Write it as dx/dy + Px = Q and apply the identical method with the variables exchanged. Board papers set this variant specifically because students who memorised one form cannot handle it.

General versus particular solutions

A general solution contains arbitrary constants — as many as the order of the equation. A particular solution is obtained by using given initial conditions to determine those constants. When a question supplies a condition such as “y = 1 when x = 0”, it is asking for a particular solution, and stopping at the general one loses the final marks. Read the question’s last line before writing your answer.

Forming a differential equation

The reverse process also appears: given a family of curves with n arbitrary constants, differentiate n times and eliminate the constants. The result is a differential equation of order n. The arithmetic is usually light; the marks are lost by differentiating too few times, or by leaving a constant in the final answer. Check explicitly that no arbitrary constant remains.

How to revise this chapter

1. Do ten questions where you only identify the type and write the first line — do not solve. This trains the decision that matters most.
2. Solve five variable separable and five homogeneous questions fully, substituting back every time.
3. Do eight linear equations, deliberately including two that are linear in x rather than y.
4. Work through the NCERT exercises and previous years’ board questions — this chapter recycles its forms closely.
5. Finish with order and degree questions, including at least one where the degree is not defined.

Because the number of question types is genuinely small, this is one of the chapters where a focused week converts directly into secure marks — which makes it a good place to build confidence before tackling heavier chapters.

FAQs

When is the degree of a differential equation not defined?

When the equation cannot be written as a polynomial in its derivatives — for example if a derivative appears inside a trigonometric, logarithmic or exponential function. An equation containing sin(dy/dx) has an order but no defined degree.

How do I know whether an equation is homogeneous?

Check whether every term has the same total degree in x and y, or equivalently whether dy/dx can be written purely as a function of y/x. If replacing x by kx and y by ky leaves the right-hand side unchanged, it is homogeneous.

Should I add a constant when finding the integrating factor?

No. Any constant introduced there appears as a multiplying factor on both sides of the final equation and cancels out. Adding it wastes time and increases the chance of an algebraic slip.

What if P and Q are functions of y instead of x?

Then the equation is linear in x rather than y. Rewrite it as dx/dy + Px = Q and apply exactly the same integrating factor method with the roles of the variables exchanged. This variant appears regularly in board papers.

Is this a scoring chapter for the board exam?

Generally yes, because the number of question types is small and they repeat closely across years. Weightage varies by board and by year, so confirm against the current official syllabus and sample paper, but the effort-to-marks ratio is usually favourable.