Differential Equations JEE Notes PDF: Download Now

Differential Equations JEE Notes: Important Concepts

Core Definitions

• Differential Equation (DE): An equation containing the derivative(s) of an unknown function. Example: $$\frac{dy}{dx}+y=0.$$
• Order: Highest derivative present. $$\frac{d^{3}y}{dx^{3}}+7y=0$$ has order $$3.$$
• Degree: Highest power of the highest order derivative, provided the equation is polynomial in the derivatives. $$\left(\frac{d^{2}y}{dx^{2}}\right)^{2}+y=0$$ has degree $$2.$$ If a DE contains $$\sin \left(\frac{dy}{dx}\right)$$, degree is not defined.
• General Solution: Family of curves with as many arbitrary constants as the order. Example: $$y=C_{1}e^{x}+C_{2}e^{-x}$$ for $$\frac{d^{2}y}{dx^{2}}=y.$$
• Particular Solution: Obtained from the general solution by assigning specific values to the constants to satisfy given initial/ boundary conditions.

Classification by Order and Linearity

Once the theory is clear, test yourself quickly from the mixed set inside JEE Questions to confirm you can identify order, degree and solution type in under 10 seconds per item.

Types of Solutions

1. Implicit: Not solved for dependent variable, e.g. $$x^{2}+y^{2}=C^{2}.$$
2. Explicit: $$y=e^{x}+C.$$
3. Singular Solution: Curve that touches every member of the general solution but is not part of it. Example: $$y^{2}=4ax$$ from the family $$\left(y-C\right)^{2}=4ax.$$

Most used formula in this section: $$n^{\text{th}}\text{ order linear DE with constant coefficients}\; \Longrightarrow\; y=C_{1}e^{m_{1}x}+C_{2}e^{m_{2}x}+\dots$$ where $$m_{i}$$ are the roots of the auxiliary equation.

First-Order Differential Equations: Types and Methods

Variable Separable Form

When $$\displaystyle \frac{dy}{dx}=g(x)\,h(y)$$, rearrange to $$\displaystyle \frac{dy}{h(y)}=g(x)\,dx$$ and integrate both sides.

Example 1

Solve $$\dfrac{dy}{dx}=x\,y^{2}.$$

Separating: $$y^{-2}\,dy=x\,dx.$$ Integrating: $$-y^{-1}=\frac{x^{2}}{2}+C.$$ Hence, $$y=-\dfrac{2}{x^{2}+C_{1}}.$$

Homogeneous Equations

For $$M(x,y)\,dx+N(x,y)\,dy=0$$ with both $$M$$ and $$N$$ homogeneous of the same degree, put $$y= vx$$ or $$x= vy$$.

Linear First-Order (Leibniz Form)

$$\displaystyle \frac{dy}{dx}+P(x)\,y=Q(x).$$ Integrating factor: $$\mu(x)=e^{\int P(x)\,dx}.$$ Solution: $$y\,\mu(x)=\int Q(x)\,\mu(x)\,dx+C.$$

Example 2

Solve $$\dfrac{dy}{dx}+2y=e^{3x}.$$

Here, $$P(x)=2 \Rightarrow \mu=e^{2x}.$$ Solution: $$y\,e^{2x}=\int e^{3x}\,e^{2x}\,dx+C=\int e^{5x}\,dx+C=\dfrac{e^{5x}}{5}+C.$$ So, $$y=\dfrac{e^{3x}}{5}+Ce^{-2x}.$$

Exact Equations

Test: $$\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}.$$ When exact, potential function $$\phi(x,y)=C$$ is obtained from $$d\phi=M\,dx+N\,dy.$$

Example 3

Check exactness and solve: $$(2xy+y^{2})\,dx+(x^{2}+2xy)\,dy=0.$$

We find $$\partial M/\partial y=2x+2y$$ and $$\partial N/\partial x=2x+2y:$$ exact. Integrate $$M$$ with respect to $$x$$: $$\phi=x^{2}y+\frac{xy^{2}}{?}$$ Wait we should integrate carefully. We'll compute: Integrate $$M=2xy+y^{2}$$ wrt $$x$$: $$\int(2xy+y^{2})dx = x^{2}y + xy^{2} + f(y).$$ Differentiate with respect to $$y$$: $$\frac{\partial\phi}{\partial y}=x^{2}+2xy+f'(y).$$ Set equal to $$N=x^{2}+2xy$$, giving $$f'(y)=0\Rightarrow f=$$ constant. Hence general solution: $$x^{2}y+xy^{2}=C.$$

Integrating Factor for Non-Exact of Special Type

If $$\dfrac{1}{N}\left(\dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x}\right)=f(x)$$ only, then $$\mu(x)=e^{\int f(x)\,dx}.$$ (Similarly for a function of $$y$$ only.)

Bernoulli’s Equation

$$\displaystyle \frac{dy}{dx}+P(x)\,y=Q(x)\,y^{n}.$$ Put $$v=y^{1-n}$$ to reduce to linear.

Clairaut and Lagrange Forms

• Clairaut: $$y=px+f(p)$$ with $$p=\dfrac{dy}{dx}.$$ General solution: the given form. Singular: eliminate $$p$$.
• Lagrange: $$y=px+f(p)$$ but with second derivative involved. Rare in JEE Main.

Higher-Order Linear Differential Equations

Auxiliary (Characteristic) Equation

For $$a_{n}\dfrac{d^{n}y}{dx^{n}}+\dots +a_{0}y=0$$ let $$y=e^{mx}.$$ Obtain $$a_{n}m^{n}+\dots +a_{0}=0.$$ Roots give the Complementary Function (CF).

Particular Integral (PI)

Standard operator method: $$\displaystyle \text{PI}=\dfrac{1}{f(D)}\;R(x)$$ where $$D=\dfrac{d}{dx}.$$

Heuristics for $$R(x)$$:

• $$e^{ax}$$: try $$Ae^{ax}.$$
• Polynomial $$P_{n}(x)$$: try polynomial of same degree.
• $$\sin bx \text{ or } \cos bx$$: assume $$A\sin bx+B\cos bx.$$ If any term clashes with CF, multiply by $$x$$ as many times as needed.

Example 4

Solve $$\dfrac{d^{2}y}{dx^{2}}-3\dfrac{dy}{dx}+2y=e^{2x}.$$

Auxiliary: $$m^{2}-3m+2=0\Rightarrow m=1,2.$$ CF: $$C_{1}e^{x}+C_{2}e^{2x}.$$ PI: RHS $$e^{2x}$$ clashes with $$e^{2x}$$ in CF, so trial $$y_{p}=Axe^{2x}.$$ Compute derivatives, substitute: $$A=1.$$ Hence, $$y=C_{1}e^{x}+C_{2}e^{2x}+xe^{2x}.$$

Cauchy-Euler (Equidimensional) Equations

General form: $$x^{n}\dfrac{d^{n}y}{dx^{n}}+\dots +a_{1}x\dfrac{dy}{dx}+a_{0}y=R(x).$$ Put $$x=e^{t},\; y=u(t)$$ or assume $$y=x^{m}.$$

To experience multi-concept combinations similar to JEE Advanced, solve the same model from previous sessions in JEE Advanced Previous Papers right after revising the PI rules.

Reduction of Order

Given one solution $$y_{1}$$ of a second-order homogeneous DE, set $$y=v\,y_{1}$$ to find a second linearly independent solution.

Method of Undetermined Coefficients vs Variation of Parameters

• Undetermined Coefficients: Faster, but only when RHS is exponential, sine/cosine or polynomial.
• Variation of Parameters: Universal but involves heavy integrals; appears in Advanced, not Main.

Application Models and JEE Typical Problems

Exponential Growth and Decay

Differential model: $$\displaystyle \frac{dN}{dt}=kN.$$ Solution: $$N=N_{0}e^{kt}.$$ Half-life $$T_{1/2}=\dfrac{\ln 2}{|k|}.$$ JEE often asks for the constant $$k$$ or time for a percentage change.

Example 5

An isotope decays at a rate proportional to the current mass. 80 g reduces to 20 g in 10 h. How much remains after 15 h?

$$N=N_{0}e^{kt},\; 20=80e^{10k}\Rightarrow e^{10k}=1/4\Rightarrow k=-\frac{\ln 4}{10}.$$ After 15 h: $$N=80e^{15k}=80\left(\dfrac14\right)^{1.5}=80\left(\dfrac{1}{8}\right)=10\text{ g}.$$ Answer: 10 g.

Newton’s Law of Cooling

$$\displaystyle \frac{dT}{dt}=-k(T-T_{s}).$$ Linear first-order, integrating gives $$T-T_{s}=(T_{0}-T_{s})e^{-kt}.$$

Orthogonal Trajectories

1. Find differential equation of given family by eliminating $$C$$.
2. Replace $$dy/dx$$ with $$-dx/dy$$ to enforce perpendicular slope.
3. Solve for the new family.

Example 6

Family: circles $$x^{2}+y^{2}=2Cy.$$ Eliminate $$C$$: $$2C=y+\dfrac{x^{2}+y^{2}}{y}\Rightarrow$$ but simpler: differentiate $$2x+2yy'=2y'.$$ Thus $$y'=\dfrac{x}{y-x}.$$ Orthogonal trajectories: replace $$y'$$ with $$-1/y'$$ giving $$y'=-\dfrac{y-x}{x}.$$ Separate: $$\dfrac{dy}{dx}=-1+\dfrac{y}{x}.$$ Linear form: $$\dfrac{dy}{dx}-\dfrac{y}{x}=-1.$$ IF $$=\mu=x^{-1}.$$ Integrate to get $$\dfrac{y}{x}= \ln x +C.$$

Simple LC Circuits (Advanced Syllabus)

For a series LC: $$L\frac{d^{2}q}{dt^{2}}+\dfrac{q}{C}=0.$$ Solution: simple harmonic: $$q=A\cos \omega t+B\sin \omega t,$$ $$\omega=\dfrac{1}{\sqrt{LC}}.$$

Practise these modelling questions along with their calculus siblings inside the test archive, then verify accuracy against official marking schemes in JEE Mains Online Coaching reports.

Important Formulas and Results at a Glance

Quick recall: $$\displaystyle \frac{dy}{dx}+P(x)\,y=Q(x)\;\Longrightarrow\; y\,e^{\int P}= \int Q\,e^{\int P}+C.$$ This single line solves half the first-order questions.

Download a printable sheet of these relations from the maths section of JEE Formula Sheets and stick it over your study desk; visual memory boosts speed.

JEE Important Points, Common Mistakes and Quick Revision

Important Points

• Weightage: 1 question (4 marks) almost every JEE Main paper; 2-step mixed concept in Advanced once every 2 years.
• Topic pairing: Linear algebraic manipulation and substitution tricks appear with DEs in Advanced. Stay fluent in log rules and partial fractions.
• Time management: First-order questions should not take more than 90 seconds when IF is obvious.

Common Student Errors

• Forgetting to add the constant of integration in separable form, losing 1 mark instantly.
• Omitting the multiplying factor $$x$$ or $$x^{2}$$ when PI clashes with CF—always check resonance.
• Applying order/degree definitions to non-polynomial DEs containing $$\sin \frac{dy}{dx}$$; remember degree undefined.
• In growth-decay, writing $$N=N_{0}e^{-kt}$$ without verifying sign of $$k$$ from data.

Rapid-Fire Revision Drill (10 min)

1. Write down the IF for $$3y'+6y=\sin 2x$$ without looking at books.
2. Solve the auxiliary equation $$D^{2}-2D+1=0,$$ state CF orally.
3. Recall half-life formula and plug in $$k=0.231$$ to get $$T_{1/2}.$
4. Derive the orthogonal trajectories of $$y^{2}=4ax$$ in your head.

Cap the session by skimming one page of solutions from JEE Mains Previous Papers to see the exact presentation style expected by NTA evaluators.