Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
How to Use This Page
This page is your complete study companion for Chapter 10 — Ordinary Differential Equations. Start with the Key Concepts table to refresh definitions, then work through the MCQs, fill-in-the-blanks, and true-or-false sections for quick practice. Use the match-the-following exercise to strengthen associations. The brief and detailed answer sections mirror the 2-mark and 5-mark questions you will face in the board exam. Finally, review the Common Mistakes and Exam Checklist before every test to avoid losing marks.
Chapter 10 — Exercise Solutions Guide
Differential equation problems are method-based. First identify the type, then apply the matching method.
Key Concepts (Quick Recap)
| Term | Definition |
|---|---|
| Differential equation | An equation involving derivatives of a function with respect to one or more variables |
| Order | The highest order of derivative appearing in the equation |
| Degree | The power of the highest order derivative when the equation is polynomial in derivatives |
| Variable separable | A method where x terms and y terms are separated on opposite sides before integration |
| Linear DE | A first-order equation of the form dy/dx + Py = Q where P and Q are functions of x |
| Integrating factor (IF) | e^(integral P dx); used to solve linear differential equations |
| Formation of DE | The process of eliminating arbitrary constants from a general solution to form a DE |
| Homogeneous DE | A DE where f(x, y) can be written as a function of y/x |
| Exact DE | A DE M dx + N dy = 0 where partial M/partial y = partial N/partial x |
| General solution | A solution containing as many arbitrary constants as the order of the DE |
| Particular solution | A solution obtained from the general solution by giving specific values to the constants |
| Initial condition | A condition that specifies the value of the function at a particular point |
| Particular integral | The solution obtained after substituting the initial condition |
Key Types
| Type | Clue |
|---|---|
| Variable separable | x terms and y terms can be separated |
| Linear differential equation | dy/dx + Py = Q |
| Formation of DE | Given general solution with constants |
| Homogeneous DE | dy/dx = f(y/x) |
I. Choose the Best Answer (MCQs)
II. Fill in the Blanks
III. True or False
1. The order of a differential equation is always a positive integer. True. (Order is the highest derivative order, which is always a positive integer.)
2. A differential equation always has a degree. False. Correction: Degree is defined only when the DE is polynomial in its derivatives. DEs containing terms like sin(dy/dx) do not have a defined degree.
3. The integrating factor of dy/dx + Py = Q is e^(integral P dx). True. (This is the standard definition of the integrating factor for a first-order linear DE.)
4. Every first-order DE can be solved by the variable separable method. False. Correction: Only DEs where the variables can be separated can be solved by the variable separable method. Others require different methods like the integrating factor method.
5. A particular solution contains arbitrary constants. False. Correction: A particular solution is obtained by substituting specific values for all arbitrary constants, so it contains no arbitrary constants.
6. The general solution of dy/dx = 0 is y = C, where C is an arbitrary constant. True. (Integrating both sides gives y = C, which is the general solution.)
7. The order of the DE formed from y = Ae^(2x) + Be^(-x) is 2. True. (Two arbitrary constants A and B require two differentiations, giving a second-order DE.)
8. In the integrating factor method, the solution is y × IF = integral(Q × IF) dx + C. True. (This is the standard formula for solving linear differential equations using the integrating factor.)
IV. Match the Following
| Column A | Column B |
|---|---|
| 1. Order of DE | (c) Highest derivative order |
| 2. Degree of DE | (f) Power of highest order derivative (when polynomial) |
| 3. Variable separable | (a) Separate x and y terms, then integrate |
| 4. Integrating factor | (d) e^(integral P dx) |
| 5. Linear DE form | (g) dy/dx + Py = Q |
| 6. General solution | (e) Contains n arbitrary constants for nth-order DE |
| 7. Particular solution | (b) Obtained using initial conditions |
| 8. Formation of DE | (h) Eliminate constants by differentiation |
Variable Separable Pattern
1. Move all y terms with dy. 2. Move all x terms with dx. 3. Integrate both sides. 4. Add constant C. 5. Simplify if required.
Linear Differential Equation Pattern
For:
Use:
Formation of Differential Equation
1. Count the number of arbitrary constants. 2. Differentiate that many times. 3. Eliminate constants. 4. Write final differential equation.
Worked Examples
Example 1 — Order and degree
State the order and degree of (d²y/dx²)³ + (dy/dx)² + y = 0.
Example 2 — Variable separable
Solve dy/dx = x/y.
Example 3 — Linear differential equation
Solve dy/dx + (1/x) y = x.
Example 4 — Formation of DE
Form the DE from y = Ae^(2x) + Be^(-x).
Example 5 — Homogeneous DE
Solve dy/dx = (x + y)/(x - y).
V. Answer Briefly (2-Mark Questions)
1. Define the order and degree of a differential equation with an example.
The order of a differential equation is the order of the highest derivative appearing in the equation. The degree is the power of the highest order derivative when the equation is expressed as a polynomial in derivatives. For example, in (d²y/dx²)³ + (dy/dx) + y = 0, the highest derivative is d²y/dx² (order 2) and its power is 3 (degree 3). Note that degree is defined only when the equation is polynomial in its derivatives.
2. What is an integrating factor? How is it used?
The integrating factor (IF) for a first-order linear differential equation dy/dx + Py = Q is defined as IF = e^(integral P dx), where P and Q are functions of x. Multiplying both sides of the DE by the IF converts the left side into an exact derivative: d/dx[y × IF] = Q × IF. Integrating both sides gives y × IF = integral(Q × IF) dx + C. This is a systematic method to solve linear DEs.
3. Distinguish between general solution and particular solution.
The general solution of a differential equation of order n contains n arbitrary constants. It represents a family of curves. A particular solution is obtained from the general solution by substituting specific values for the arbitrary constants using initial conditions. For example, dy/dx = 2x has general solution y = x² + C. If y(0) = 5, then C = 5, giving the particular solution y = x² + 5.
4. How do you form a differential equation from a given family of curves?
To form a differential equation from a family of curves with n arbitrary constants, differentiate the equation n times to get n + 1 equations (including the original). Then eliminate all n arbitrary constants from these equations to obtain the required differential equation of order n. For example, y = mx + c has two constants; differentiating twice gives d²y/dx² = 0, which is the required second-order DE.
5. Solve the differential equation dy/dx = e^(x - y).
This is a variable separable DE. Writing dy/dx = e^(x-y) = e^x × e^(-y). Separating variables: e^y dy = e^x dx. Integrating both sides: e^y = e^x + C. This is the general solution. If an initial condition is given, we can find C to get the particular solution.
VI. Answer in Detail (5-Mark Questions)
1. Explain the variable separable method for solving first-order differential equations and solve dy/dx = (1 + y²)/(1 + x²).
The variable separable method is used when a first-order DE can be written in the form f(y)dy = g(x)dx, i.e., the terms involving y and dy can be separated from the terms involving x and dx. The steps are: (1) rewrite the DE so that all y terms are on one side with dy and all x terms are on the other side with dx; (2) integrate both sides; (3) add the constant of integration C; (4) simplify to obtain the general solution. For the given DE dy/dx = (1 + y²)/(1 + x²), we separate: dy/(1 + y²) = dx/(1 + x²). Integrating both sides: tan⁻¹(y) = tan⁻¹(x) + C. This can also be written as tan⁻¹(y) - tan⁻¹(x) = C, or y = (x + tan C)/(1 - x tan C) using the tan⁻¹ subtraction formula. If an initial condition is given, say y(0) = 1, then tan⁻¹(1) - tan⁻¹(0) = C gives C = pi/4. The particular solution is tan⁻¹(y) - tan⁻¹(x) = pi/4, which simplifies to y = (x + 1)/(1 - x).
2. Explain the method of solving a linear differential equation using the integrating factor. Solve dy/dx + 2y/x = x².
A linear first-order differential equation has the form dy/dx + Py = Q, where P and Q are functions of x (or constants). The integrating factor method involves: (1) identify P and Q from the standard form; (2) compute IF = e^(integral P dx); (3) multiply both sides of the DE by the IF; (4) the left side becomes d/dx[y × IF]; (5) integrate both sides with respect to x; (6) solve for y. For dy/dx + 2y/x = x²: Here P = 2/x and Q = x². IF = e^(integral 2/x dx) = e^(2 ln x) = x². Multiplying both sides by x²: x²(dy/dx) + 2xy = x⁴. The left side is d/dx[yx²]. Integrating: yx² = integral(x⁴) dx = x⁵/5 + C. Therefore y = x³/5 + C/x². This is the general solution. Verification: substituting back into the original DE confirms correctness.
3. Form the differential equation by eliminating the arbitrary constants from y = Ae^(3x) + Be^(-2x). Also find the order and degree of the resulting DE.
The given equation y = Ae^(3x) + Be^(-2x) contains two arbitrary constants A and B. To eliminate them, we need to differentiate twice. First derivative: dy/dx = 3Ae^(3x) - 2Be^(-2x). Second derivative: d²y/dx² = 9Ae^(3x) + 4Be^(-2x). Now we have three equations: (i) y = Ae^(3x) + Be^(-2x), (ii) y' = 3Ae^(3x) - 2Be^(-2x), (iii) y'' = 9Ae^(3x) + 4Be^(-2x). From (i) and (ii): 2y + y' = 5Ae^(3x) and y' - 3y = -5Be^(-2x). From (ii) and (iii): 2y' + y'' = 15Ae^(3x) and y'' - 2y' = 10Be^(-2x). Alternatively, a more direct approach: multiply (i) by 5: 5y = 5Ae^(3x) + 5Be^(-2x). Then (5y) + (y') = 5Ae^(3x)(1 + 3) + 5Be^(-2x)(1 - 2) leads to elimination. A cleaner method: from (i), (ii), (iii), we can eliminate A and B systematically to get d²y/dx² - dy/dx - 6y = 0. The resulting DE has order 2 (highest derivative is d²y/dx²) and degree 1 (the power of the highest derivative is 1).
4. Solve the initial value problem dy/dx = y cos x, given y(0) = 3.
| This is a variable separable DE. Separating variables: dy/y = cos x dx. Integrating both sides: ln | y | = sin x + C₁. Exponentiating: y = Ce^(sin x), where C = e^(C₁). This is the general solution. Applying the initial condition y(0) = 3: 3 = Ce^(sin 0) = Ce⁰ = C. So C = 3. The particular solution is y = 3e^(sin x). We can verify: dy/dx = 3e^(sin x) × cos x = y cos x, which matches the original DE. At x = 0, y = 3e^(sin 0) = 3e⁰ = 3, confirming the initial condition is satisfied. |
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5. What is a homogeneous differential equation? Solve dy/dx = (x² + y²)/(2x²).
| A homogeneous differential equation is one where dy/dx can be written as a function of y/x alone. That is, if we substitute y = vx, the right side becomes a function of v only. The substitution y = vx transforms the DE into a separable equation in v and x. For dy/dx = (x² + y²)/(2x²), we write dy/dx = (1 + (y/x)²)/2 = (1 + v²)/2 where v = y/x. Since y = vx, dy/dx = v + x(dv/dx). So v + x(dv/dx) = (1 + v²)/2. This gives x(dv/dx) = (1 + v²)/2 - v = (1 - 2v + v²)/2 = (1 - v)²/2. Separating: 2dv/(1-v)² = dx/x. Integrating: 2/(1-v) = ln | x | + C. Substituting v = y/x: 2/(1-y/x) = ln | x | + C, which simplifies to 2x/(x-y) = ln | x | + C. |
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Common Mistakes to Avoid
1. ❌ Forgetting to add the constant of integration C after integrating. ✅ Always add "+ C" when integrating an indefinite integral in a differential equation problem. The constant is essential for the general solution.
2. ❌ Confusing order and degree of a differential equation. ✅ Order is the highest derivative order (a positive integer). Degree is the power of that highest derivative (only defined for polynomial DEs).
3. ❌ Not writing the DE in standard form dy/dx + Py = Q before computing the integrating factor. ✅ Always rearrange to dy/dx + Py = Q first. If the equation is given as dy/dx = Q - Py, rewrite it before identifying P.
4. ❌ Using the wrong integrating factor formula. ✅ The IF is e^(integral P dx), where P is the coefficient of y in the standard form dy/dx + Py = Q. Compute P first, then integrate.
5. ❌ Trying to use variable separable method on a linear DE that is not separable. ✅ First check if variables can be separated. If not, check if the DE is linear (form dy/dx + Py = Q) and use the integrating factor method.
6. ❌ Not differentiating enough times when forming a DE from a family of curves. ✅ The number of differentiations must equal the number of arbitrary constants. Two constants require two differentiations.
7. ❌ Making integration errors in the integrating factor or in the final solution. ✅ Double-check all integrations. Common errors include forgetting chain rule adjustments and sign errors.
8. ❌ Forgetting initial conditions lead to particular solutions, not general solutions. ✅ After finding the general solution, substitute the initial condition values to determine C and write the particular solution.
Exam Checklist
- Identify the type of DE (variable separable, linear, homogeneous) before choosing a method.
- For order and degree questions, find the highest derivative order first, then check its power.
- In the variable separable method, separate all y terms with dy and all x terms with dx before integrating.
- Always add the constant of integration C after integrating.
- For linear DEs, first write in standard form dy/dx + Py = Q, then compute the IF = e^(integral P dx).
- Multiply the entire DE by the IF before integrating.
- For formation of DE problems, count the number of arbitrary constants and differentiate that many times.
- When solving initial value problems, substitute the given values after finding the general solution.
- Verify your answer by differentiating and substituting back into the original DE.
- Express the final answer in the simplest form possible.
- For homogeneous DEs, always use the substitution y = vx (or x = vy) and show the step where dv/dx appears.