Differential Equations – KCET Mathematics Study Notes
(For Karnataka Common Entrance Test – Core Mathematics)
Differential equations are equations that involve an unknown function and its derivatives.
They are the basic language of modelling physical, biological, and economic systems.
1. Simple Explanation
| Term | Meaning |
|---|---|
| Differential Equation (DE) | An equation that contains one or more derivatives of an unknown function. |
| Order | Highest derivative that appears (e.g., first‑order if the highest derivative is (y')). |
| Degree | Power of the highest derivative if the DE is polynomial in derivatives. |
| Linear DE | A DE that is linear in the unknown function and its derivatives. |
| Homogeneous DE | All terms involve the unknown function (no “free” terms). |
| Particular Solution | A specific solution that satisfies the non‑homogeneous part. |
| General Solution | Sum of the complementary (homogeneous) solution and a particular solution. |
Why study DE?
They describe how a variable changes with respect to another (time, space, etc.) – e.g., population growth, cooling, oscillations.
2. Key Points & Solution Methods
| Order | Type | Typical Form | Solution Strategy | Key Formula |
|---|---|---|---|---|
| 1 | Separable | (\displaystyle \frac{dy}{dx}=f(x)g(y)) | Separate variables → integrate both sides | (\displaystyle \int \frac{1}{g(y)}dy=\int f(x),dx) |
| 1 | Linear | (\displaystyle \frac{dy}{dx}+P(x)y=Q(x)) | Integrating factor (\mu(x)=e^{\int P(x)dx}) | (y,\mu = \int Q(x)\mu,dx + C) |
| 1 | Bernoulli | (\displaystyle \frac{dy}{dx}+P(x)y=Q(x)y^n) | Substitute (v=y^{1-n}) → linear in (v) | (v' + (1-n)P(x)v=(1-n)Q(x)) |
| 1 | Exact | (M(x,y)+N(x,y)y'=0) | Check (\partial M/\partial y=\partial N/\partial x) → find potential (F(x,y)=C) | (F(x,y)=C) |
| 2 | Linear homogeneous with constant coefficients | (ay''+by'+cy=0) | Characteristic equation (ar^2+br+c=0) → roots (r_1,r_2) | If distinct: (y=C_1e^{r_1x}+C_2e^{r_2x}) |
| 2 | Non‑homogeneous | (ay''+by'+cy=g(x)) | Find complementary (y_c) + particular (y_p) | (y=y_c+y_p) |
| Method of Undetermined Coefficients | (g(x)) is polynomial, (e^{kx}), (\sin kx), (\cos kx) | Guess form of (y_p) with constants → solve | ||
| Variation of Parameters | General (g(x)) | Use (y_p=u_1y_1+u_2y_2) with (u'_1, u'_2) from Wronskian | (u'_1=-\frac{y_2g}{W},; u'_2=\frac{y_1g}{W}) |
Quick Checklist for Each DE
- Identify the order & type.
- Check if it’s linear or separable.
- For linear first‑order: compute integrating factor.
- For Bernoulli: make substitution to linearise.
- For exact: verify exactness condition; if not exact, find integrating factor.
- For second‑order linear: solve characteristic equation; then add particular solution.
3. Worked Example
Problem
Solve the first‑order linear differential equation
[
\frac{dy}{dx}+2y=\sin x,\qquad y(0)=1.
]
Solution Steps
-
Standard form
[ \frac{dy}{dx}+P(x)y=Q(x)\quad\text{with }P(x)=2,\ Q(x)=\sin x. ] -
Integrating factor
[ \mu(x)=e^{\int P(x),dx}=e^{\int 2,dx}=e^{2x}. ] -
Multiply the DE by (\mu)
[ e^{2x}\frac{dy}{dx}+2e^{2x}y =e^{2x}\sin x. ] Left side is (\displaystyle \frac