What traumatic event does Zitkala-Sa describe at the school?

The forcible cutting of her long hair

Why was the cutting of her hair so humiliating to Zitkala-Sa?

Among her people, shingled hair was worn by cowards and mourners

Define the order of a differential equation.

It is the order of the highest-order derivative occurring in the equation.

Define the degree of a differential equation.

When the equation is a polynomial in its derivatives, the degree is the highest power of the highest-order derivative.

How many arbitrary constants are in a general solution?

As many as the order of the differential equation.

Describe the variables-separable method.

For \(\dfrac{dy}{dx}=g(x)h(y)\), write \(\dfrac{dy}{h(y)}=g(x)\,dx\) and integrate both sides.

How is a homogeneous differential equation solved?

If \(\dfrac{dy}{dx}=F(y/x)\), substitute \(y=vx\), so \(\dfrac{dy}{dx}=v+x\dfrac{dv}{dx}\), reducing it to a separable equation in \(v\) and \(x\).

Write the standard form of a first-order linear equation.

\(\dfrac{dy}{dx}+Py=Q\), where \(P\) and \(Q\) are functions of \(x\) only.

State the integrating factor for \(\dfrac{dy}{dx}+Py=Q\).

The integrating factor is \(\mathrm{I.F.}=e^{\int P\,dx}\).

Write the solution of a linear differential equation.

\(y\cdot(\mathrm{I.F.})=\int Q\cdot(\mathrm{I.F.})\,dx+C\).

Find the order and degree of \(\dfrac{d^2y}{dx^2}+\left(\dfrac{dy}{dx}\right)^3+y=0\).

The order is \(2\) and the degree is \(1\).

What distinguishes a particular solution?

It is obtained from the general solution by assigning specific values to the arbitrary constants using given initial or boundary conditions.

Differential Equations

Class 12 Mathematics · Mathematics Part II · Chapter 3

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CBSE Class 12 · Mathematics

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Summary

A differential equation relates a function to its derivatives. Its order is the order of the highest derivative present, and its degree (when defined) is the power of that highest-order derivative once the equation is a polynomial in derivatives. A solution containing as many arbitrary constants as the order is the general solution; fixing the constants from initial conditions gives a particular solution. This chapter covers the formation of differential equations and three methods of solving first-order, first-degree equations: variables separable, where \(\dfrac{dy}{dx}=g(x)h(y)\) is rearranged and integrated; homogeneous equations, solved by the substitution \(y=vx\); and linear equations \(\dfrac{dy}{dx}+Py=Q\), solved using the integrating factor \(e^{\int P\,dx}\). Differential equations model growth, decay and many physical processes.

Order and degreeFormation of differential equationsVariables separable methodHomogeneous differential equationsLinear differential equations

Key terms

Order

The order of the highest derivative in the equation.

Degree

The power of the highest-order derivative when the equation is polynomial in derivatives.

General solution

A solution with as many arbitrary constants as the order.

Variables separable

Form \(\dfrac{dy}{dx}=g(x)h(y)\), solved by separating and integrating.

Homogeneous equation

Solved by the substitution \(y=vx\).

Integrating factor

\(e^{\int P\,dx}\) for the linear equation \(\dfrac{dy}{dx}+Py=Q\).

Extra questions & answers

Try one from this chapter

The order of a differential equation is:

Flashcards & interactive practice