Chapter 9 · Calculus

Differential Equations

5 topics 105 practice questions

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Chapter Overview

A differential equation links an unknown function with its derivatives: \(\frac{dy}{dx}=ky\) says "the rate of growth is proportional to the size". Such equations model population growth, cooling, radioactive decay, interest and motion. In this chapter you will classify differential equations by order and degree, check solutions, and solve three important first-order types: variables separable, homogeneous and linear.

Board focus: order/degree MCQs, one separable or homogeneous equation and one linear equation (often with an initial condition).

Topics

1 Order and Degree 17 questions
2 General and Particular Solutions 13 questions
3 Variables Separable Method 25 questions
4 Homogeneous Differential Equations 19 questions
5 Linear Differential Equations 31 questions

Key Concepts

Order = highest derivative present; degree = power of that derivative when the equation is a polynomial in derivatives.
Separable: \(\frac{dy}{dx}=f(x)g(y)\Rightarrow\int\frac{dy}{g(y)}=\int f(x)dx\).
Homogeneous: \(\frac{dy}{dx}=F\left(\frac yx\right)\); put \(y=vx\).
Linear: \(\frac{dy}{dx}+Py=Q\); integrating factor \(e^{\int P\,dx}\), solution \(y\cdot\text{IF}=\int Q\cdot\text{IF}\,dx+C\).

Formulas

Exponential growth/decay

Differential Equations · Variables Separable Method

\[\frac{dP}{dt}=kP\;\Rightarrow\;P=P_0e^{kt}\]

Homogeneous substitution

Differential Equations · Homogeneous Differential Equations

\[y=vx,\quad \frac{dy}{dx}=v+x\frac{dv}{dx}\]

Linear equation

Differential Equations · Linear Differential Equations

\[\frac{dy}{dx}+Py=Q:\quad \text{IF}=e^{\int P\,dx},\quad y\,\text{IF}=\int Q\,\text{IF}\,dx+C\]

Key Points

When degree is not defined

Differential Equations · Order and Degree

If a derivative appears inside sin, cos, e, log (e.g. \(y''+e^{y'}=0\)), the equation is not a polynomial in derivatives and its degree is not defined.

Number of constants

Differential Equations · General and Particular Solutions

The general solution of an nth-order equation has exactly n arbitrary constants.

Common Mistakes

Degree with radicals

Differential Equations · Order and Degree

For \(\left(1+(y')^2\right)^{3/2}=y''\), square both sides first: the degree is 2, not \(\frac23\) or undefined.

Losing the constant

Differential Equations · Variables Separable Method

After \(\log|y|=x^2+C\), write \(y=Ae^{x^2}\) – the constant multiplies, it does not add.

Standard form first

Differential Equations · Linear Differential Equations

For \(x\frac{dy}{dx}+2y=x^2\), divide by \(x\) first: \(P=\frac2x\), \(\text{IF}=x^2\) – not \(e^{\int2\,dx}\).

Solved Examples

Homogeneous equation

Differential Equations · Homogeneous Differential Equations

Solve \(\frac{dy}{dx}=\frac{y}{x}+\frac{x}{y}\).

\(y=vx\): \(x\frac{dv}{dx}=\frac1v\Rightarrow v\,dv=\frac{dx}x\Rightarrow\frac{v^2}2=\log|x|+C\Rightarrow y^2=2x^2(\log|x|+C)\).

Linear equation

Differential Equations · Linear Differential Equations

Solve \(\frac{dy}{dx}+y=e^{x}\).

\(\text{IF}=e^x\). \(ye^x=\int e^{2x}dx=\frac{e^{2x}}2+C\Rightarrow y=\frac{e^x}2+Ce^{-x}\).

Practice & Tests

Chapter Practice

Untimed mixed questions from all topics. Answers are revealed after you submit.

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Chapter Test

Timed test – about 15 questions in 30 minutes.

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Differential Equations – Chapter Test

A board-style chapter test drawn fresh from the question bank each time: MCQs, assertion-reason, numericals, written answers you self-check against model solutions, and a case study.

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