Applied Mathematics · Class 12 Commerce
Ch 5Differential Equations and Modeling — Class 12 Applied Mathematics, concept-first.
This concept map shows how the chapter fits together. It covers the order and degree of a differential equation, its general and particular solutions, the formation of a differential equation from a family of curves, and mathematical modeling with differential equations — growth and decay, population growth, compound i…
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Order and Degree of Differential Equations
A differential equation is an equation that involves derivatives — rates of change. When you see something like
Most relevant Q&A
- Find the order and degree (if defined) of the following differential equations: (i) $\frac{dy}{dx}=ky$, where $k$ is a scalar (ii) $\left(\f…Preview
- Form a differential equation representing the family of parabolas having vertex at origin and axis along positive direction of y-axis.Free
- Form the differential equation of the family of hyperbolas having foci on $x$-axis and Centre at origin.Preview
- Solve the differential equation: $y\log y\,dx-x\,dy=0$Free
- Determine the order and degree (if defined) of the differential equation: $x\frac{dy}{dx}+2y=x^2$, $x\ne 0$Free
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Concept Map
This concept map shows how the chapter fits together. It covers the order and degree of a differential equation, its general and particular solutions, the formation of a differential equation from a f…
Differential Equations
A differential equation is an equation that involves a derivative of one variable with respect to another — not just the variable itself, but how fast it is changing.
+−Exercise 1i5 questions
- Q1Determine the order and degree (if defined) of the differential equation: $x\frac{dy}{dx}+2y=x^2$, $x\ne 0$Free
- Q2Determine the order and degree (if defined) of the differential equation: $\frac{dy}{dx}+e^y=0$Free
- Q3Determine the order and degree (if defined) of the differential equation: $\frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$Preview
- Q4Determine the order and degree (if defined) of the differential equation: $\left(\frac{dy}{dx}\right)^4+3y\left(\frac{d^2y}{dx^2}\right)=0$Preview
- Q5Determine the order and degree (if defined) of the differential equation: $(y''')^2+(y'')^3+(y')^4+y^5=0$, where $y'=\frac{dy}{dx}$, $y''=\f…Preview
Order of a Differential Equation
The order of a differential equation is the order of the highest derivative of the dependent variable (with respect to the independent variable) that appears in the equation.
Degree of a Differential Equation
The degree of a differential equation is defined only when the equation is a polynomial equation in its derivatives — that is, every derivative appearing in it occurs raised to a positive integral pow…
General and Particular Solutions of a Differential Equation
In earlier classes, solving an equation such as meant finding the real numbers that make both sides equal.
+−Exercise 2i7 questions
- Q1Verify that given function (explicit or implicit) is a solution of the corresponding differential equation: $y=ae^{-x}$ ; $\frac{dy}{dx}+y=0…Free
- Q2Verify that given function (explicit or implicit) is a solution of the corresponding differential equation: $y=\sqrt{1+x^2}$ ; $\frac{dy}{dx…Free
- Q3Verify that given function (explicit or implicit) is a solution of the corresponding differential equation: $xy=\log y+c$ ; $\frac{dy}{dx}=\…Free
- Q4Verify that given function (explicit or implicit) is a solution of the corresponding differential equation: $ax^2+by^2=1$ ; $x(yy_2+y_1^2)=y…Preview
- Q5Verify that given function (explicit or implicit) is a solution of the corresponding differential equation: $y=(a+bx)e^{2x}$ ; $y_2-4y_1+4y=…Preview
- Q6Verify that given function (explicit or implicit) is a solution of the corresponding differential equation: $x^2=2y^2\log y$ ; $(x^2+y^2)\fr…Preview
- Q7Verify that the function, $y=ke^x-1$ is a solution of the differential equation $\frac{dy}{dx}=y+1$. Also determine the value of the constan…Preview
General Solution
The general solution of a differential equation is a function of the independent and dependent variables that also contains independent arbitrary constants, equal in number to the order of the differe…
Particular Solution
A particular solution is any solution obtained from the general solution by substituting specific numerical values in place of its arbitrary constant(s).
+−Worked Examplesi4 questions
- Example 2Verify that the function $y=ae^{bx}$ is a solution of the differential equation $\frac{d^2y}{dx^2}-b^2y=0$Free
- Example 3Verify that $y=ce^{-x^3}$ is the solution of the differential equation $\frac{dy}{dx}+3x^2y=0$. Also determine the solution curve of the giv…Free
- Example 4Verify that $y=\frac{1}{x}-\log x$ is a solution of the differential equation $x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}-y=\log x$Preview
- Example 5Show that $y^2=4ax$ is a solution of the differential equation, $y=x\frac{dy}{dx}+a\frac{dx}{dy}$Preview
Formation of a Differential Equation
So far you've learned what a differential equation is and what its solution looks like. Now turn the process around: given a family of curves — infinitely many curves sharing the same shape but differ…
+−Exercise 3i6 questions
- Q1Form the differential equation not containing the arbitrary constants and satisfied by the equation $x^2-y^2=a^2$, where $a$ is an arbitrary…Free
- Q2Find the differential equation of the family of circles having centre at origin.Free
- Q3Form the differential equation of the family of circles having centre on $y$-axis and passing through origin.Preview
- Q4Form the differential equation representing the family of curves $y=e^{2x}(a+bx)$, where $a,b$ are arbitrary constants.Preview
- Q5Find the differential equation representing the parabolas having their vertices at origin and foci on positive direction of x-axis.Preview
- Q6Form the differential equation of the family of ellipses having their foci on $x$-axis and centre at the origin.Preview
Steps to Form a Differential Equation
When a family of curves depends on independent parameters, forming its differential equation is a systematic elimination process:
+−Worked Examplesi3 questions
- Example 6Form a differential equation representing the family of parabolas having vertex at origin and axis along positive direction of y-axis.Free
- Example 7Form a differential equation representing the family of curves given by $y=ae^{bx}$, where $a,b$ are arbitrary constantsPreview
- Example 8Form the differential equation of the family of hyperbolas having foci on $x$-axis and Centre at origin.Preview
Solving Simple Differential Equation
12 QOnce a differential equation is set up, the next task is to actually solve it. This section covers the most basic first-order, first-degree equations — ones where the variables can be separated onto o…
+−Worked Examplesi4 questions
- Example 9Find the general solution of the differential equation $\frac{dy}{dx}+y=1$, $(y\ne 1)$Free
- Example 10Solve the differential equation: $y\log y\,dx-x\,dy=0$Free
- Example 11Solve the differential equation: $\frac{dy}{dx}=e^{x+y}+x^2e^y$Preview
- Example 12Find the particular solution of the differential equation $\frac{dy}{dx}=x(2\log x+1)$, given that $y=0$ when $x=2$Preview
+−Exercise 4i8 questions
- Q1Find the general solution of the differential equation: $\frac{dy}{dx}=(e^x+1)y$Free
- Q2Find the general solution of the differential equation: $x^5\frac{dy}{dx}=-y^5$Free
- Q3Find the general solution of the differential equation: $\frac{dy}{dx}=\frac{x+1}{2-y}$Free
- Q4Find the general solution of the differential equation: $x(e^{2y}-1)dy+(x^2-1)e^y\,dx=0$Preview
- Q5Find the general solution of the differential equation: $e^x\sqrt{1-y^2}\,dx+\frac{y}{x}\,dy=0$Preview
- Q6Find the equation of the curve passing through the point $(1, -1)$ whose differential equation is $xy\frac{dy}{dx}=(x+2)(y+2)$Preview
- Q7Solve $(x+1)\frac{dy}{dx}=2xy$, given that $y(2)=3$Preview
- Q8Find the particular solution of the differential equation $\log\left(\frac{dy}{dx}\right)=3x+4y$, given that $y=0$, when $x=0$.Preview
Differential Equations and Mathematical Modeling
Differential equations aren't just solved for their own sake — they are the language used to build mathematical models of real-world change.
+−Exercise 5i11 questions
- Q1Find an exponential growth model, $y=y_0e^{kt}$ that satisfies the stated conditions: i. $y_0=1$ and doubling time $t=5$ years. ii. $y(0)=5$…Free
- Q2Gaurav deposited ₹5000 in an account paying 3% interest compounded continuously for 5 years. i. Find the total amount at the end of 5 years.…Free
- Q3In a certain culture of bacteria, the number of bacteria increased 5 times in 10 hours. How long did it take for the number of bacteria to d…Free
- Q4The amount of oil pumped from one of the wells decreases at the continuous rate of 10% per year. When will the wells output fall to one-four…Preview
- Q5A cup of tea with temperature 95°C is placed in a room with a constant temperature of 21°C. How many minutes will it take to reach a tempera…Preview
- Q6A cake is removed from an oven at 250°F and left to cool at room temperature which is 70°F. After 30 minutes the temperature of the cake is…Preview
- Q7Radium decomposes at a rate proportional to the amount present. If half the original amount disappears in 1600 years, find the percentage lo…Preview
- Q8Half-life of radioactive carbon-14 is 5700 years. A certain bone was observed to contain 75% of carbon-14 as compared to what is present in…Preview
- Q9If 600 grams of a radioactive substance are present initially and 3 years later only 300 grams remain. How much of the substance will be pre…Preview
- Q10The space vehicles are supplied power from nuclear energy derived from radioactive isotopes. The output of the radioactive power supply for…Preview
- Q11Use the exponential growth model to show that the time it takes for a population to double (i.e., from an initial number A to 2A) is given b…Preview
The Process of Mathematical Modeling
Turning a real-world situation into usable mathematics — and the mathematics back into a real-world answer — follows the same three-stage process, regardless of the application:
Growth and Decay Models
Many quantities in nature and finance change at a rate proportional to their own current size — more of the quantity present means a faster rate of change.
Population Growth
Consider a population of individuals — human, insect, or bacterial — at time , with a constant birth rate and a constant death rate.
Compound Interest
Suppose an amount is deposited in a bank account at an annual interest rate , and — instead of interest being credited once a year or once a quarter — it is compounded continuously, i.e., added to the…
Newton's Law of Cooling
When a hot object is left in a cooler room — or a cold drink is left on a warm table — its temperature doesn't change at a constant rate; it changes fastest when the gap between the object and its sur…
Carbon Dating
Carbon dating is a technique used to estimate the age of the remains of plants and animals. Living organisms continuously exchange carbon with their environment, which keeps the proportion of radioact…
Drug Assimilation Into the Blood
When a pill is swallowed, its ingredients don't reach the bloodstream instantly — they first dissolve in the gastrointestinal tract (GI tract) and diffuse from there into the blood, from where the bod…