📈 \( \frac{d^2y}{dx^2} + \frac{dy}{dx} + y = 0 \)
📈 Chapter 15: Differential Equations | NEB Mathematics Notes Class 12
Based on Latest Syllabus 2080 | Order, Degree, Variable Separable, Homogeneous, Linear DE
✅ Updated according to latest syllabus of 2080 | Complete notes with 15 solved problems
📈 Differential Equations — This chapter covers order and degree of differential equations, formation of differential equations, variable separable method, homogeneous differential equations, linear differential equations, and exact differential equations. The notes have been updated according to the latest syllabus of 2080. Now you don’t need to go anywhere searching for the notes of this chapter because we are here to serve you.
📚 Exercises
📖 Definition of Differential Equation
A differential equation is an equation that involves an unknown function and its derivatives. It represents a relationship between the function and its rate of change.
Example: \(\frac{dy}{dx} = f(x, y)\) where \(y\) is dependent variable and \(x\) is independent variable.
Example: \(\frac{dy}{dx} = f(x, y)\) where \(y\) is dependent variable and \(x\) is independent variable.
📐 Order and Degree of Differential Equations
Order: The highest order derivative present in the differential equation.
Example: \(\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0\) → Order = 2
Example: \(\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0\) → Order = 2
Degree: The power of the highest order derivative (when equation is polynomial in derivatives).
Example: \(\left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} = 0\) → Degree = 2
Example: \(\left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} = 0\) → Degree = 2
🔄 Formation of Differential Equations
A differential equation can be formed by eliminating arbitrary constants from a given equation.
Rule: If the equation contains \(n\) arbitrary constants, differentiate \(n\) times and eliminate the constants.
Rule: If the equation contains \(n\) arbitrary constants, differentiate \(n\) times and eliminate the constants.
📌 General and Particular Solutions
General Solution: Contains arbitrary constants equal to the order of the DE. Represents a family of solutions.
Particular Solution: Obtained by giving specific values to arbitrary constants using initial conditions.
🔧 Methods of Solving Differential Equations
1. Variable Separable: \(\frac{dy}{dx} = f(x)g(y)\) ⇒ \(\int \frac{dy}{g(y)} = \int f(x) dx\)
2. Homogeneous DE: \(\frac{dy}{dx} = f\left(\frac{y}{x}\right)\) ⇒ Substitute \(y = vx\)
3. Linear DE: \(\frac{dy}{dx} + Py = Q\) ⇒ Integrating factor = \(e^{\int P dx}\)
4. Exact DE: \(M dx + N dy = 0\) with \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\)
📝 Solved Problems (15 Important Questions)
Q1 Find the order and degree of \(\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 + y = 0\).
Highest derivative: \(\frac{d^2y}{dx^2}\) → Order = 2
Power of highest derivative: 1 → Degree = 1
Power of highest derivative: 1 → Degree = 1
✅ Order = 2, Degree = 1
Q2 Find the order and degree of \(\left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 = 0\).
Highest derivative: \(\frac{d^2y}{dx^2}\) → Order = 2
Power of highest derivative: 3 → Degree = 3
Power of highest derivative: 3 → Degree = 3
✅ Order = 2, Degree = 3
Q3 Solve \(\frac{dy}{dx} = \frac{x}{y}\).
Variable separable: \(y dy = x dx\)
\(\int y dy = \int x dx\) ⇒ \(\frac{y^2}{2} = \frac{x^2}{2} + C\) ⇒ \(y^2 – x^2 = C\)
\(\int y dy = \int x dx\) ⇒ \(\frac{y^2}{2} = \frac{x^2}{2} + C\) ⇒ \(y^2 – x^2 = C\)
✅ \(y^2 – x^2 = C\)
Q4 Solve \(\frac{dy}{dx} = \frac{1 + y^2}{1 + x^2}\).
Separate variables: \(\frac{dy}{1+y^2} = \frac{dx}{1+x^2}\)
\(\int \frac{dy}{1+y^2} = \int \frac{dx}{1+x^2}\) ⇒ \(\tan^{-1}y = \tan^{-1}x + C\)
\(\int \frac{dy}{1+y^2} = \int \frac{dx}{1+x^2}\) ⇒ \(\tan^{-1}y = \tan^{-1}x + C\)
✅ \(\tan^{-1}y – \tan^{-1}x = C\)
Q5 Solve \(\frac{dy}{dx} = e^{x-y}\).
\(\frac{dy}{dx} = e^x \cdot e^{-y}\) ⇒ \(e^y dy = e^x dx\)
\(\int e^y dy = \int e^x dx\) ⇒ \(e^y = e^x + C\)
\(\int e^y dy = \int e^x dx\) ⇒ \(e^y = e^x + C\)
✅ \(e^y – e^x = C\)
Q6 Solve \(\frac{dy}{dx} = \frac{x + y}{x}\).
\(\frac{dy}{dx} = 1 + \frac{y}{x}\). Homogeneous, let \(y = vx\) ⇒ \(\frac{dy}{dx} = v + x\frac{dv}{dx}\)
\(v + x\frac{dv}{dx} = 1 + v\) ⇒ \(x\frac{dv}{dx} = 1\) ⇒ \(dv = \frac{dx}{x}\)
\(v = \ln|x| + C\) ⇒ \(\frac{y}{x} = \ln|x| + C\) ⇒ \(y = x\ln|x| + Cx\)
\(v + x\frac{dv}{dx} = 1 + v\) ⇒ \(x\frac{dv}{dx} = 1\) ⇒ \(dv = \frac{dx}{x}\)
\(v = \ln|x| + C\) ⇒ \(\frac{y}{x} = \ln|x| + C\) ⇒ \(y = x\ln|x| + Cx\)
✅ \(y = x\ln|x| + Cx\)
Q7 Solve \(\frac{dy}{dx} = \frac{y}{x} + \tan\frac{y}{x}\).
Let \(y = vx\), \(\frac{dy}{dx} = v + x\frac{dv}{dx}\)
\(v + x\frac{dv}{dx} = v + \tan v\) ⇒ \(x\frac{dv}{dx} = \tan v\)
\(\cot v dv = \frac{dx}{x}\) ⇒ \(\ln|\sin v| = \ln|x| + \ln C\)
\(\sin v = Cx\) ⇒ \(\sin\left(\frac{y}{x}\right) = Cx\)
\(v + x\frac{dv}{dx} = v + \tan v\) ⇒ \(x\frac{dv}{dx} = \tan v\)
\(\cot v dv = \frac{dx}{x}\) ⇒ \(\ln|\sin v| = \ln|x| + \ln C\)
\(\sin v = Cx\) ⇒ \(\sin\left(\frac{y}{x}\right) = Cx\)
✅ \(\sin\left(\frac{y}{x}\right) = Cx\)
Q8 Solve \(\frac{dy}{dx} + 2y = e^x\).
Linear DE of form \(\frac{dy}{dx} + Py = Q\) with \(P = 2\), \(Q = e^x\)
Integrating factor \(I.F. = e^{\int 2 dx} = e^{2x}\)
Solution: \(y \cdot e^{2x} = \int e^x \cdot e^{2x} dx = \int e^{3x} dx = \frac{e^{3x}}{3} + C\)
\(y = \frac{e^x}{3} + C e^{-2x}\)
Integrating factor \(I.F. = e^{\int 2 dx} = e^{2x}\)
Solution: \(y \cdot e^{2x} = \int e^x \cdot e^{2x} dx = \int e^{3x} dx = \frac{e^{3x}}{3} + C\)
\(y = \frac{e^x}{3} + C e^{-2x}\)
✅ \(y = \frac{e^x}{3} + Ce^{-2x}\)
Q9 Solve \(\frac{dy}{dx} + y\cot x = \csc x\).
\(P = \cot x\), \(Q = \csc x\)
\(I.F. = e^{\int \cot x dx} = e^{\ln|\sin x|} = \sin x\)
\(y \sin x = \int \csc x \cdot \sin x dx = \int 1 dx = x + C\)
\(I.F. = e^{\int \cot x dx} = e^{\ln|\sin x|} = \sin x\)
\(y \sin x = \int \csc x \cdot \sin x dx = \int 1 dx = x + C\)
✅ \(y \sin x = x + C\)
Q10 Solve \(x\frac{dy}{dx} – y = x^2\).
Rewrite: \(\frac{dy}{dx} – \frac{y}{x} = x\)
\(P = -\frac{1}{x}\), \(I.F. = e^{\int -\frac{1}{x}dx} = e^{-\ln|x|} = \frac{1}{x}\)
\(y \cdot \frac{1}{x} = \int x \cdot \frac{1}{x} dx = \int 1 dx = x + C\)
\(y = x^2 + Cx\)
\(P = -\frac{1}{x}\), \(I.F. = e^{\int -\frac{1}{x}dx} = e^{-\ln|x|} = \frac{1}{x}\)
\(y \cdot \frac{1}{x} = \int x \cdot \frac{1}{x} dx = \int 1 dx = x + C\)
\(y = x^2 + Cx\)
✅ \(y = x^2 + Cx\)
Q11 Form the differential equation from \(y = A e^{2x} + B e^{-3x}\).
\(y’ = 2A e^{2x} – 3B e^{-3x}\)
\(y” = 4A e^{2x} + 9B e^{-3x}\)
Eliminating A and B: \(y” – y’ – 6y = 0\)
\(y” = 4A e^{2x} + 9B e^{-3x}\)
Eliminating A and B: \(y” – y’ – 6y = 0\)
✅ \(\frac{d^2y}{dx^2} – \frac{dy}{dx} – 6y = 0\)
Q12 Solve \((x^2 + 1)\frac{dy}{dx} = 2xy\).
Separate: \(\frac{dy}{y} = \frac{2x}{x^2+1} dx\)
\(\int \frac{dy}{y} = \int \frac{2x}{x^2+1} dx\) ⇒ \(\ln|y| = \ln|x^2+1| + C\)
\(y = C(x^2+1)\)
\(\int \frac{dy}{y} = \int \frac{2x}{x^2+1} dx\) ⇒ \(\ln|y| = \ln|x^2+1| + C\)
\(y = C(x^2+1)\)
✅ \(y = C(x^2+1)\)
Q13 Solve \(\frac{dy}{dx} = \sqrt{\frac{1-y^2}{1-x^2}}\).
Separate: \(\frac{dy}{\sqrt{1-y^2}} = \frac{dx}{\sqrt{1-x^2}}\)
\(\int \frac{dy}{\sqrt{1-y^2}} = \int \frac{dx}{\sqrt{1-x^2}}\) ⇒ \(\sin^{-1}y = \sin^{-1}x + C\)
\(\int \frac{dy}{\sqrt{1-y^2}} = \int \frac{dx}{\sqrt{1-x^2}}\) ⇒ \(\sin^{-1}y = \sin^{-1}x + C\)
✅ \(\sin^{-1}y – \sin^{-1}x = C\)
Q14 Solve \(\frac{dy}{dx} = \frac{y}{x} + \frac{x}{y}\).
Let \(y = vx\), then \(\frac{dy}{dx} = v + x\frac{dv}{dx} = v + \frac{1}{v}\)
\(x\frac{dv}{dx} = \frac{1}{v}\) ⇒ \(v dv = \frac{dx}{x}\)
\(\frac{v^2}{2} = \ln|x| + C\) ⇒ \(\frac{y^2}{2x^2} = \ln|x| + C\)
\(x\frac{dv}{dx} = \frac{1}{v}\) ⇒ \(v dv = \frac{dx}{x}\)
\(\frac{v^2}{2} = \ln|x| + C\) ⇒ \(\frac{y^2}{2x^2} = \ln|x| + C\)
✅ \(y^2 = 2x^2\ln|x| + Cx^2\)
Q15 Solve \(\frac{dy}{dx} + \frac{y}{x} = \frac{1}{x^2}\).
Linear DE: \(P = \frac{1}{x}\), \(Q = \frac{1}{x^2}\)
\(I.F. = e^{\int \frac{1}{x}dx} = e^{\ln|x|} = x\)
\(y \cdot x = \int \frac{1}{x^2} \cdot x dx = \int \frac{1}{x} dx = \ln|x| + C\)
\(y = \frac{\ln|x| + C}{x}\)
\(I.F. = e^{\int \frac{1}{x}dx} = e^{\ln|x|} = x\)
\(y \cdot x = \int \frac{1}{x^2} \cdot x dx = \int \frac{1}{x} dx = \ln|x| + C\)
\(y = \frac{\ln|x| + C}{x}\)
✅ \(y = \frac{\ln|x| + C}{x}\)
✏️ Practice Questions
P1 Find order and degree of \(\frac{d^2y}{dx^2} + \sin\left(\frac{dy}{dx}\right) = 0\).
✅ Answer: Order = 2, Degree = 1 (since sin term is not polynomial → degree not defined)
P2 Solve \(\frac{dy}{dx} = \frac{1 – x^2}{1 – y^2}\).
✅ Answer: \(y – \frac{y^3}{3} = x – \frac{x^3}{3} + C\)
P3 Solve \(\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}\) (Homogeneous).
✅ Answer: \(y^2 = x^2 \ln|x| + Cx^2\)
P4 Solve \(\frac{dy}{dx} + 3y = e^{-2x}\).
✅ Answer: \(y = e^{-2x} + Ce^{-3x}\)
P5 Form the DE from \(y = A\cos x + B\sin x\).
✅ Answer: \(\frac{d^2y}{dx^2} + y = 0\)
📋 Multiple Choice Questions (HSEB Pattern)
MCQ 1 The order of \(\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 = 0\) is:
A) 1 B) 2 C) 3 D) 0
✅ Answer: B) 2
MCQ 2 The degree of \(\left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} = 0\) is:
A) 1 B) 2 C) 3 D) Not defined
✅ Answer: B) 2
MCQ 3 The integrating factor for \(\frac{dy}{dx} + y = e^x\) is:
A) \(e^x\) B) \(e^{-x}\) C) \(x\) D) \(\ln x\)
✅ Answer: A) \(e^x\)
MCQ 4 The differential equation \(\frac{dy}{dx} = \frac{y}{x}\) is:
A) Linear B) Homogeneous C) Variable separable D) All of these
✅ Answer: D) All of these
MCQ 5 The solution of \(\frac{dy}{dx} = 2y\) is:
A) \(y = e^{2x} + C\) B) \(y = Ce^{2x}\) C) \(y = 2e^{x} + C\) D) \(y = e^{x/2} + C\)
✅ Answer: B) \(y = Ce^{2x}\)
MCQ 6 The general solution of \(\frac{dy}{dx} = \frac{x}{y}\) is:
A) \(x^2 + y^2 = C\) B) \(x^2 – y^2 = C\) C) \(y^2 – x^2 = C\) D) \(x^2 + y = C\)
✅ Answer: C) \(y^2 – x^2 = C\)
MCQ 7 The number of arbitrary constants in the general solution of a second order DE is:
A) 1 B) 2 C) 3 D) 0
✅ Answer: B) 2
MCQ 8 The integrating factor for \(\frac{dy}{dx} + \frac{y}{x} = x\) is:
A) \(x^2\) B) \(x\) C) \(\ln x\) D) \(e^x\)
✅ Answer: B) \(x\)
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This PDF provides the solutions of every question from the exercises of class 12 differential equations chapter. If you want the notes of other exercises then you can choose the exercise from the button given above.
This PDF provides the solutions of every question from the exercises of class 12 differential equations chapter. If you want the notes of other exercises then you can choose the exercise from the button given above.
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📌 Key Differential Equations Summary:
• Order: Highest derivative in the equation
• Degree: Power of highest order derivative
• Variable Separable: \(\frac{dy}{dx} = f(x)g(y)\) ⇒ \(\int \frac{dy}{g(y)} = \int f(x) dx\)
• Homogeneous: Substitute \(y = vx\), then separate variables
• Linear: \(\frac{dy}{dx} + Py = Q\), I.F. = \(e^{\int P dx}\), Solution: \(y \cdot I.F. = \int Q \cdot I.F. dx + C\)
• Formation of DE: Eliminate arbitrary constants by differentiation
• Order: Highest derivative in the equation
• Degree: Power of highest order derivative
• Variable Separable: \(\frac{dy}{dx} = f(x)g(y)\) ⇒ \(\int \frac{dy}{g(y)} = \int f(x) dx\)
• Homogeneous: Substitute \(y = vx\), then separate variables
• Linear: \(\frac{dy}{dx} + Py = Q\), I.F. = \(e^{\int P dx}\), Solution: \(y \cdot I.F. = \int Q \cdot I.F. dx + C\)
• Formation of DE: Eliminate arbitrary constants by differentiation