Order and degree, general and particular solutions, and solving first order first degree equations: variables separable, homogeneous and linear. Our own step-by-step solutions to Exercises 9.1 to 9.5 and the Miscellaneous Exercise, one page per exercise, set out for step marks.
Each exercise has its own page with every question solved step by step. Pick the one you are on.
What it tests. Order = the highest derivative present. Degree = the power of that highest derivative once the equation is a polynomial in all the derivatives; if a derivative sits inside sin, log, e and so on, the degree is not defined.
What it tests. Differentiate the given function (implicitly if needed) as many times as the order of the equation, substitute into the left side, and simplify to the right side. For an implicit solution, find \(y'\) by implicit differentiation and use the original relation to simplify.
What it tests. If \(\tfrac{dy}{dx} = g(x)h(y)\), write \(\tfrac{dy}{h(y)} = g(x)\,dx\) and integrate both sides (one constant is enough). For a particular solution, put the given point in to find C. Growth problems: \(\tfrac{dP}{dt} = kP\) gives \(P = P_0e^{kt}\).
What it tests. If \(\tfrac{dy}{dx} = F\left(\tfrac{y}{x}\right)\) (every term of the same degree), put \(y = vx\), \(\tfrac{dy}{dx} = v + x\tfrac{dv}{dx}\); the equation separates in v and x. For \(\tfrac{dx}{dy} = h\left(\tfrac{x}{y}\right)\) put \(x = vy\). Replace v by \(\tfrac{y}{x}\) at the end.
What it tests. For \(\tfrac{dy}{dx} + Py = Q\) (P, Q functions of x): integrating factor \(\text{IF} = e^{\int P\,dx}\), and \(y \cdot \text{IF} = \int Q \cdot \text{IF}\,dx + C\). If the equation is linear in x instead, use \(\tfrac{dx}{dy} + P_1x = Q_1\) with \(\text{IF} = e^{\int P_1\,dy}\).
What it tests. Mixed practice: identify the type first (separable, homogeneous, linear in y or in x), then use the matching method; for 'verify' questions, differentiate and substitute.
Differential Equations has 40 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.
Also useful: free MCQs and case studies for Differential Equations · Class 12 formula sheet · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan
Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.