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EAPCET Mathematics · Intermediate second year · Paper IIB

Differential equations

Differential Equations covers the order and degree of an equation, forming equations from families of curves, and solving separable, homogeneous and linear equations.

Subject
Mathematics
Taught in
Intermediate second year · Paper IIB
Mathematics in the paper
80 of 160 questions

Key formulas and results

  • Order is the highest derivative; degree is its power once the equation is polynomial in derivatives
  • A family with nn arbitrary constants gives a differential equation of order nn
  • Homogeneous equations: substitute y=vxy = vx
  • dydx+Py=Q\dfrac{dy}{dx} + Py = Q: integrating factor e∫P dxe^{\int P\,dx}, and y⋅IF=∫Q⋅IF dx+Cy \cdot \text{IF} = \displaystyle\int Q \cdot \text{IF}\,dx + C
  • Variable separable: dydx=g(x) h(y)\dfrac{dy}{dx} = g(x)\,h(y) gives ∫dyh(y)=∫g(x) dx+C\displaystyle\int \dfrac{dy}{h(y)} = \int g(x)\,dx + C
  • dydx=f(ax+by+c)\dfrac{dy}{dx} = f(ax + by + c): put ax+by+c=tax + by + c = t, so dydx=dtdx−ab\dfrac{dy}{dx} = \dfrac{\frac{dt}{dx} - a}{b}, which separates
  • Reducible to homogeneous: for dydx=ax+by+cAx+By+D\dfrac{dy}{dx} = \dfrac{ax + by + c}{Ax + By + D}, put x=X+hx = X + h, y=Y+ky = Y + k with ah+bk+c=0ah + bk + c = 0 and Ah+Bk+D=0Ah + Bk + D = 0
  • Bernoulli: dydx+Py=Qyn\dfrac{dy}{dx} + Py = Qy^{n}; put t=y1−nt = y^{1-n} to get the linear equation dtdx+(1−n)Pt=(1−n)Q\dfrac{dt}{dx} + (1 - n)Pt = (1 - n)Q

What this chapter covers

  • Order and degree
  • Formation of differential equations
  • Variables separable
  • Homogeneous equations
  • Linear first-order equations

How it is usually asked

  • Find the order and degree
  • Form the equation of a family of curves
  • Solve a linear or homogeneous equation

Understanding Differential equations

A differential equation relates a function to its derivatives. Its order is the order of the highest derivative present, and its degree is the power of that highest derivative once the equation is written as a polynomial in the derivatives; if that is impossible, as with sin(dy/dx), the degree is not defined. A family of curves with n independent arbitrary constants is described by a differential equation of order n.

To form that equation, differentiate the family’s equation as many times as there are constants, and eliminate the constants. To solve an equation, first identify its type. If the variables can be separated as f(y) dy = g(x) dx, integrate both sides. If it is homogeneous, with dy/dx a function of y/x, substitute y = vx, which turns it into a separable equation in v and x.

A linear equation dy/dx + Py = Q, with P and Q functions of x, is solved by the integrating factor IF = e^(∫P dx): multiplying through makes the left side the derivative of y · IF, so y · IF = ∫ Q · IF dx + C. EAPCET questions ask for order and degree, the equation of a family of curves, or the general solution of one of these three types.

Worked example

EAPCET-style question

The general solution of dydx+yx=x2\dfrac{dy}{dx} + \dfrac{y}{x} = x^{2} (x>0)(x > 0) is

  1. (1)xy=x44+Cxy = \dfrac{x^{4}}{4} + C
  2. (2)y=x34+Cy = \dfrac{x^{3}}{4} + C
  3. (3)xy=x33+Cxy = \dfrac{x^{3}}{3} + C
  4. (4)yx=x22+C\dfrac{y}{x} = \dfrac{x^{2}}{2} + C

Solution

  1. The equation is linear with P=1xP = \dfrac{1}{x} and Q=x2Q = x^{2}.
  2. IF=e∫(1/x) dx=elog⁡x=x\text{IF} = e^{\int (1/x)\,dx} = e^{\log x} = x.
  3. So y⋅x=∫x2⋅x dx=∫x3 dx=x44+Cy \cdot x = \displaystyle\int x^{2} \cdot x\,dx = \int x^{3}\,dx = \dfrac{x^{4}}{4} + C.
  4. The solution is xy=x44+Cxy = \dfrac{x^{4}}{4} + C.

Answer: xy=x44+Cxy = \dfrac{x^{4}}{4} + C

Shortcuts and results to remember

  • The degree is not defined when a derivative appears inside sin, log, an exponential and so on, after the equation has been cleared of radicals and fractions.
  • Example: (y′′′)5−(y′′)10+5y (y(4))3−7y=0(y''')^{5} - (y'')^{10} + 5y\,(y^{(4)})^{3} - 7y = 0 has order 4 (highest derivative y(4)y^{(4)}) and degree 3 (its power).
  • In dydx+Py=Q\dfrac{dy}{dx} + Py = Q, PP and QQ must be functions of xx only; if the equation is linear in xx instead, use dxdy+P1x=Q1\dfrac{dx}{dy} + P_{1}x = Q_{1} with IF=e∫P1 dy\text{IF} = e^{\int P_{1}\,dy}.

More practice questions

Practice question 1

The general solution of xlog⁡x dydx+y=2log⁡xx\log x\,\dfrac{dy}{dx} + y = 2\log x is

  1. (1)ylog⁡x=(log⁡x)2+Cy\log x = (\log x)^{2} + C
  2. (2)y=log⁡x+Cy = \log x + C
  3. (3)ylog⁡x=2log⁡x+Cy\log x = 2\log x + C
  4. (4)y=(log⁡x)2+Cy = (\log x)^{2} + C

Solution

  1. Divide by xlog⁡xx\log x: dydx+yxlog⁡x=2x\dfrac{dy}{dx} + \dfrac{y}{x\log x} = \dfrac{2}{x}
  2. IF=e∫dxxlog⁡x=elog⁡(log⁡x)=log⁡x\text{IF} = e^{\int \frac{dx}{x\log x}} = e^{\log(\log x)} = \log x
  3. ylog⁡x=∫2xlog⁡x dxy\log x = \displaystyle\int \dfrac{2}{x}\log x\,dx
  4. Put t=log⁡xt = \log x: ∫2t dt=t2=(log⁡x)2\displaystyle\int 2t\,dt = t^{2} = (\log x)^{2}
  5. Solution: ylog⁡x=(log⁡x)2+Cy\log x = (\log x)^{2} + C

Answer: ylog⁡x=(log⁡x)2+Cy\log x = (\log x)^{2} + C

Practice question 2

The general solution of dydx=(x+y+2)2\dfrac{dy}{dx} = (x + y + 2)^{2} is

  1. (1)tan⁡−1(x+y+2)=x+C\tan^{-1}(x + y + 2) = x + C
  2. (2)tan⁡−1(x+y+2)=y+C\tan^{-1}(x + y + 2) = y + C
  3. (3)x+y+2=tan⁡x+Cx + y + 2 = \tan x + C
  4. (4)(x+y+2)3=3x+C(x + y + 2)^{3} = 3x + C

Solution

  1. Put t=x+y+2t = x + y + 2, so dtdx=1+dydx\dfrac{dt}{dx} = 1 + \dfrac{dy}{dx}
  2. The equation becomes dtdx−1=t2\dfrac{dt}{dx} - 1 = t^{2}, i.e. dtdx=1+t2\dfrac{dt}{dx} = 1 + t^{2}
  3. Separate: ∫dt1+t2=∫dx\displaystyle\int \dfrac{dt}{1 + t^{2}} = \int dx
  4. tan⁡−1t=x+C\tan^{-1}t = x + C, so tan⁡−1(x+y+2)=x+C\tan^{-1}(x + y + 2) = x + C

Answer: tan⁡−1(x+y+2)=x+C\tan^{-1}(x + y + 2) = x + C

Common mistakes to avoid

  • Reading P before making the coefficient of dy/dx equal to 1; divide the whole equation by that coefficient first.
  • Giving a degree to an equation like sin(dy/dx) + y = 0; it is not a polynomial in the derivative, so its degree is not defined.
  • Adding the constant C only after dividing by the integrating factor; write y · IF = ∫ Q · IF dx + C and then divide, so C is divided too.
  • Counting constants that can be combined, such as c₁ + c₂ or c₁e^(c₂), as separate; the order equals the number of independent constants.

How to prepare it

Identify the type of equation before trying to solve it — the method follows directly from the type.

Once the formulas feel familiar, test them the way the exam does: practise Differential equations questions chapter-wise, then attempt a full previous year paper to see whether it holds up under a 3-hour timer.

Related Mathematics chapters

See every chapter in the EAPCET Mathematics syllabus.

Practise Differential equations chapter-wise

Chapter-wise quizzes show exactly which topics in this chapter you still get wrong — before a full paper does.

Frequently asked questions

Is Differential equations part of the EAPCET Mathematics syllabus?
Yes. Differential equations is taught in Intermediate second year · Paper IIB Mathematics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.
How should I prepare Differential equations for EAPCET?
Identify the type of equation before trying to solve it — the method follows directly from the type.
How many Mathematics questions are there in EAPCET?
80 of the 160 questions in the Engineering paper are Mathematics, spread across 37 chapters.
What mistakes do students make in Differential equations?
Reading P before making the coefficient of dy/dx equal to 1; divide the whole equation by that coefficient first. Giving a degree to an equation like sin(dy/dx) + y = 0; it is not a polynomial in the derivative, so its degree is not defined. Adding the constant C only after dividing by the integrating factor; write y · IF = ∫ Q · IF dx + C and then divide, so C is divided too. Counting constants that can be combined, such as c₁ + c₂ or c₁e^(c₂), as separate; the order equals the number of independent constants.