Class 12 Maths Differential Equations: Step-by-Step Board Method
This post explains how a Class 12 student should solve and write differential equation answers in the board exam.
You will get a clear method, one worked example, and a weekly routine you can follow at home.
Why this chapter worries parents in Dwarka and Gurgaon
A differential equation is an equation that has a derivative in it, such as dy/dx. Solving it means finding the function y that satisfies the equation. The chapter in the NCERT book looks short, but it joins two big topics: differentiation and integration. A weak spot in either one shows up here.
Parents in Dwarka and Gurgaon often describe the same pattern. The child understood in class but cannot start the question. The final answer is right, but the marks are low. The student forgets the constant, or forgets to use the given condition at the end.
These are habit problems, not intelligence problems. They can be fixed with a method and regular practice.
Step 1: identify the type before you do anything
Many lost marks come from picking the wrong method. Spend the first few seconds naming the type. For CBSE Class 12, these are the first-order equations your child must handle.
| Type | How to recognise it | First move |
|---|---|---|
| Variable separable | You can move all y terms with dy and all x terms with dx | Separate, then integrate both sides |
| Homogeneous | dy/dx = f(x, y), and every term has the same total power of x and y | Put y = vx, so dy/dx = v + x(dv/dx) |
| Linear first order | Fits the form dy/dx + P·y = Q, where P and Q depend on x only (or are constants) | Find the integrating factor, e∫P dx |
Tell your child to write the type as the first line of the answer, in plain words, for example: this is a linear equation of the first order. It costs a few seconds. It also shows the examiner that the method was chosen on purpose.
The integrating factor is a multiplier. When you multiply the whole linear equation by it, the left side becomes the derivative of a single product. That is why the method works.
Two basics that must be solid first
Order and degree. The order is the highest derivative in the equation. The degree is the power of that highest derivative, once the equation is free of roots and fractions in the derivatives. Take (d2y/dx2)3 + (dy/dx)2 + y = 0. Its order is 2 and its degree is 3.
General and particular solution. The general solution has an arbitrary constant, usually C. A particular solution has a fixed value of C, found from a given condition such as y = 2 when x = 0. Many students stop at the general solution. If the question gives a condition, the last step is to find C.
Worked example: a linear equation with a condition
Problem. Solve dy/dx + 2y = e−x, given that y = 2 when x = 0.
- Type. The equation fits the form dy/dx + P·y = Q, with P = 2 and Q = e−x. It is a linear equation.
- Integrating factor. The integrating factor is e∫2 dx = e2x.
- Write the solution form. y · e2x = ∫ e−x · e2x dx.
- Simplify and integrate. The right side becomes ∫ ex dx = ex + C. So y · e2x = ex + C.
- General solution. y = e−x + C e−2x.
- Use the condition. Put x = 0 and y = 2. Then 2 = 1 + C, so C = 1.
- Particular solution. y = e−x + e−2x.
Quick check: at x = 0 the answer gives 1 + 1 = 2, which matches the condition.
Notice what this answer shows: the type, the integrating factor, the integral, the constant and the final result. Each is a separate line, and each can earn its own mark.
How to write the answer so marks are not lost
- One step per line. Do not combine integration and simplification in a single line.
- Write the formula first. Write the integrating factor formula before you put values in. Then substitute.
- Add the constant at the right moment. Write + C as soon as you integrate. Adding it later often causes wrong signs.
- Handle logarithms carefully. In separable questions, the constant can be written as log C to keep the answer tidy. Say so in one line and stay consistent.
- Finish with a closing line. State the required solution clearly at the end. Do not leave the answer hanging in the middle of the page.
- Verify when time allows. Differentiate your answer once and see if it fits the original equation.
For homogeneous equations, students often forget to go back to x and y at the end. After integrating in v, replace v with y/x. Make this a written step in the notebook, not something to remember.
For separable equations, check the separation before integrating. If a term like (1 + y2) sits in the wrong place, the integral becomes harder than it should be. Take a few seconds to confirm that dy has only y terms and dx has only x terms.
Common mistakes we see, and the fix
| Mistake | Fix |
|---|---|
| Wrong type chosen | Name the type in the first line, using the table above |
| Integration slips | The equation is not the problem. Revise basic integrals first |
| Given condition ignored | Underline the condition when you read the question |
| Sign errors from messy work | Write neatly, one step per line, so you can read your own work |
A weekly routine for the home
Differential equations are best learned in short, regular sessions. One long session just before the exam works poorly. Here is a simple plan for a student who is doing this chapter now.
- Day 1: Revise order, degree and forming an equation from a family of curves.
- Day 2: Practise only variable separable questions. Write each one fully, as in the exam.
- Day 3: Practise only homogeneous questions. Focus on the substitution y = vx.
- Day 4: Practise only linear equations. Find the integrating factor first, every time.
- Day 5: Do a mixed set. The student names the type of each question before solving it.
- Weekend: Redo the questions that went wrong, without looking at the earlier solution. Then compare.
Keep a small mistake page at the back of the notebook. Each wrong step goes there in one line. Reading it before every session is more useful than solving ten new questions.
Parents do not need to know the chapter. Ask your child to explain, in their own words, why they chose the method for one question. If they can say it clearly, the understanding is there.
How a home tutor helps with this chapter
This chapter rewards someone who watches the student write. A tutor sitting beside the child can see exactly where the method breaks: the wrong type, a missed constant, or an integration slip. In a group, that detail is easy to miss.
Great Home Tutors offers one-to-one tuition at your home. It is home tuition only, with no coaching centre and no group batch. You tell us the class, subject, board and your locality. A person from our team calls you to understand the student. We then tell you plainly whether a tutor is verified by us or not yet. The first class is a chance to see the fit before anything is fixed.
Whichever route you choose, ask the tutor to check the written answer line by line, and not only the final result.
Key takeaways
- Name the type first: separable, homogeneous or linear. Then apply its fixed method.
- Write one step per line and end with a clear closing line.
- If a condition is given, find C. The general solution alone is incomplete.
- Study this chapter in short daily sessions, with a mistake page for errors.
- Weak integration is the hidden cause of many wrong answers here, so revise it early.
Frequently asked questions
What is the first thing to do in a differential equation question?
Identify the type. Check whether the variables can be separated, whether the equation is homogeneous, or whether it fits the linear form. The method follows from the type.
Do I lose marks if I skip the constant C?
It is a real risk. The general solution needs the arbitrary constant. A particular solution needs its value from the given condition. Write both clearly.
Is this chapter only about formulas?
No. The formulas are few. The real work is recognising the type and integrating correctly. That is why practice with written steps matters more than memorising.
How long should my child practise each day?
Short and regular is better. A focused session on one type each day, followed by a mixed set, works better than one long session. Adjust the time to your child's school load.
Can a home tutor in Dwarka or Gurgaon help with Class 12 Maths?
Yes. Great Home Tutors offers one-to-one home tuition for Class 12 Maths in Dwarka and Gurgaon. You share the class, subject, board and locality, and we match a tutor. No result is guaranteed, and the first class lets you judge the fit.
What to do next. Try it yourself on the page below.
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