Balbharati Maharashtra State Board Std 12 Commerce Statistics Part 1 Digest Pdf Chapter 8 Differential Equation and Applications Ex 8.1 Questions and Answers.

## Maharashtra State Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.1

Question 1.

Determine the order and degree of each of the following differential equations:

(i) \(\frac{d^{2} x}{d t^{2}}+\left(\frac{d x}{d t}\right)^{2}+8=0\)

Solution:

The given D.E. is \(\frac{d^{2} x}{d t^{2}}+\left(\frac{d x}{d t}\right)^{2}+8=0\)

This D.E. has highest order derivative \(\frac{d^{2} x}{d t^{2}}\) with power 1.

∴ the given D.E. is of order 2 and degree 1.

(ii) \(\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+\left(\frac{d y}{d x}\right)^{2}=a^{x}\)

Solution:

The given D.E. is \(\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+\left(\frac{d y}{d x}\right)^{2}=a^{x}\)

This D.E. has highest order derivative \(\frac{d^{2} y}{d x^{2}}\) with power 2.

∴ the given D.E. is of order 2 and degree 2.

(iii) \(\frac{d^{4} y}{d x^{4}}+\left[1+\left(\frac{d y}{d x}\right)^{2}\right]^{3}\)

Solution:

The given D.E. is \(\frac{d^{4} y}{d x^{4}}+\left[1+\left(\frac{d y}{d x}\right)^{2}\right]^{3}\)

This D.E. has highest order derivative \(\frac{d^{4} y}{d x^{4}}\) with power 1.

∴ the given D.E. is of order 4 and degree 1.

(iv) (y'”)^{2} + 2(y”)^{2} + 6y’ + 7y = 0

Solution:

The given D.E. is (y”‘)^{2} + 2(y”)^{2} + 6y’ + 7y = 0

This can be written as \(\left(\frac{d^{3} y}{d x^{3}}\right)^{2}+2\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+6 \frac{d y}{d x}+7 y=0\)

This D.E. has highest order derivative \(\frac{d^{3} y}{d x^{3}}\) with power 2.

∴ the given D.E. is of order 3 and degree 2.

(v) \(\sqrt{1+\frac{1}{\left(\frac{d y}{d x}\right)^{2}}}=\left(\frac{d y}{d x}\right)^{3 / 2}\)

Solution:

The given D.E. is \(\sqrt{1+\frac{1}{\left(\frac{d y}{d x}\right)^{2}}}=\left(\frac{d y}{d x}\right)^{3 / 2}\)

On squaring both sides, we get

\(1+\frac{1}{\left(\frac{d y}{d x}\right)^{2}}=\left(\frac{d y}{d x}\right)^{3}\)

∴ \(\left(\frac{d y}{d x}\right)^{2}+1=\left(\frac{d y}{d x}\right)^{5}\)

This D.E. has highest order derivative \(\frac{d y}{d x}\) with power 5.

∴ the given D.E. is of order 1 and degree 5.

(vi) \(\frac{d y}{d x}=7 \frac{d^{2} y}{d x^{2}}\)

Solution:

The given D.E. is \(\frac{d y}{d x}=7 \frac{d^{2} y}{d x^{2}}\)

This D.E. has highest order derivative \(\frac{d^{2} y}{d x^{2}}\) with power 1.

∴ the given D.E. is of order 2 and degree 1.

(vii) \(\left(\frac{d^{3} y}{d x^{3}}\right)^{1 / 6}=9\)

Solution:

The given D.E. is \(\left(\frac{d^{3} y}{d x^{3}}\right)^{1 / 6}=9\)

i.e., \(\frac{d^{3} y}{d x^{3}}=9^{6}\)

This D.E. has highest order derivative \(\frac{d^{3} y}{d x^{3}}\) with power 1.

∴ the given D.E. is of order 3 and degree 1.

Question 2.

In each of the following examples, verify that the given function is a solution of the corresponding differential equation:

Solution:

(i) xy = log y + k

Differentiating w.r.t. x, we get

Hence, xy = log y + k is a solution of the D.E. y'(1 – xy) = y^{2}.

(ii) y = x^{n}

Differentiating twice w.r.t. x, we get

This shows that y = x^{n} is a solution of the D.E.

\(x^{2} \frac{d^{2} y}{d x^{2}}-n x \frac{d y}{d x}+n y=0\)

(iii) y = e^{x}

Differentiating w.r.t. x, we get

\(\frac{d y}{d x}\) = e^{x} = y

Hence, y = ex is a solution of the D.E. \(\frac{d y}{d x}\) = y.

(iv) y = 1 – log x

Differentiating w.r.t. x, we get

Hence, y = 1 – log x is a solution of the D.E.

\(x^{2} \frac{d^{2} y}{d x^{2}}=1\)

(v) y = ae^{x} + be^{-x}

Differentiating w.r.t. x, we get

\(\frac{d y}{d x}\) = a(e^{x}) + b(-e^{-x}) = ae^{x} – be^{-x}

Differentiating again w.r.t. x, we get

\(\frac{d^{2} y}{d x^{2}}\) = a(e^{x}) – b(-e^{-x})

= ae^{x} + be^{-x}

= y

Hence, y = ae^{x} + be^{-x} is a solution of the D.E. \(\frac{d^{2} y}{d x^{2}}\) = y.

(vi) ax^{2} + by^{2} = 5

Differentiating w.r.t. x, we get

Hence, ax^{2} + by^{2} = 5 is a solution of the D.E.

\(x y \frac{d^{2} y}{d x^{2}}+x\left(\frac{d y}{d x}\right)^{2}=y\left(\frac{d y}{d x}\right)\)