Class 12 Mathematics Chapter 9: Differential Equations
CBSE 2026–27 | Detailed NCERT-aligned study resource
This chapter covers definition, order and degree, general and particular solutions, separation of variables, homogeneous first-order equations and linear differential equations. citeturn0search11
1. What Is a Differential Equation?
An equation containing derivatives of a dependent variable is a differential equation. The order is the highest derivative present. The degree is the power of that highest-order derivative after the equation is polynomial in derivatives.
Example: d²y/dx²+3dy/dx+2y=0 has order 2 and degree 1.
2. General and Particular Solutions
A general solution contains arbitrary constants. A condition such as y(0)=2 can determine the constant and produce a particular solution.
3. Separation of Variables
When variables can be separated, rearrange the equation into a function of y times dy equal to a function of x times dx, then integrate both sides.
Example: dy/dx=2xy. Then dy/y=2x dx. Integrating gives ln|y|=x²+C, so y=Ceˣ².
4. Homogeneous First-Order Equations
For an equation reducible to a function of y/x, use y=vx. Then dy/dx=v+x dv/dx. The resulting equation is usually separable.
↓
dy/dx=v+x(dv/dx)
↓
separate v and x
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integrate and replace v=y/x
5. Linear Differential Equation
For dy/dx+Py=Q, the integrating factor is IF=e^(∫Pdx). Multiplying the equation by IF converts the left side into d(y·IF)/dx.
Formula: y·IF=∫Q·IF dx+C.
Worked Example
Solve dy/dx+y=x. Here P=1, Q=x and IF=eˣ. Thus d(yeˣ)/dx=xeˣ. Integrating gives yeˣ=eˣ(x−1)+C, so y=x−1+Ce⁻ˣ.
6. Linear Equation in dx/dy
CBSE also includes dx/dy+px=q, where p and q are functions of y or constants. Treat y as the independent variable and use IF=e^(∫pdy).
7. Formation of a Differential Equation
If a family of curves contains arbitrary constants, differentiate enough times to eliminate those constants. The resulting equation is the differential equation of the family.
Example: y=ceˣ. Differentiating gives dy/dx=ceˣ=y. Therefore the differential equation is dy/dx−y=0.
8. Method Selection Guide
| Form | First method to consider |
|---|---|
| Variables can be separated | Separation of variables |
| Homogeneous in x and y | Put y=vx |
| dy/dx+Py=Q | Integrating factor |
| dx/dy+px=q | Integrating factor with y as independent variable |
9. Common Mistakes
- Confusing order with degree.
- Forgetting arbitrary constants.
- Using y=vx but failing to transform dy/dx.
- Using the wrong variable in an integrating factor.
- Not applying the given initial condition to the general solution.
10. Practice Set
- Find order and degree of five differential equations.
- Form a differential equation from a one-parameter family.
- Solve a separable equation with an initial condition.
- Solve a homogeneous first-order equation using y=vx.
- Solve dy/dx+3y=6.
- Solve an equation written in dx/dy using its integrating factor.