Exercise 10.1 - Chapter 10 Ordinary Differential Equations 12th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Ex $10.1$
Question 1.
For each of the following differential equations, determine its order, degree (if exists)
(i) $\frac{d y}{d x}+x y=\cot x$
Solution:
Order $=1$
Degree $=1$
(ii) $\left(\frac{d^3 y}{d x^3}\right)^{\frac{2}{3}}-3 \frac{d^2 y}{d x^2}+5 \frac{d y}{d x}+4=0$
Solution:
$
\left(\frac{d^3 y}{d x^3}\right)^{\frac{2}{3}}=3 \frac{d^2 y}{d x^2}-5 \frac{d y}{d x}-4
$
Raising the power to ' 3 ' on both the sides
$
\begin{aligned}
\left(\frac{d^3 y}{d x^3}\right)^2 & =\left(3 \frac{d^2 y}{d x^2}-5 \frac{d y}{d x}-4\right)^3 \\
\text { Order } & =3, \text { degree }=2
\end{aligned}
$

(iii) $\left(\frac{d^2 y}{d x^2}\right)^2+\left(\frac{d y}{d x}\right)^2=x \sin \left(\frac{d^2 y}{d x^2}\right)$
Solution:
Highest derivative is $\frac{d^2 y}{d x^2}$
$\therefore$ Order $=2$
Degree does not exist as $\frac{d^2 y}{d x^2}$ cannot be separated from $\left(\frac{d^2 y}{d x^2}\right)$.

(iv) $\sqrt{\frac{d y}{d x}}-4 \frac{d y}{d x}-7 x=0$
Solution:
$
\sqrt{\frac{d y}{d x}}=4 \frac{d y}{d x}+7 x
$
Squaring on both sides
$
\frac{d y}{d x}=\left(4 \frac{d y}{d x}+7 x\right)^2
$
Order $=1$, degree $=2$.

(v)
$
y\left(\frac{d y}{d x}\right)=\frac{x}{\left(\frac{d y}{d x}\right)+\left(\frac{d y}{d x}\right)^3}
$
Solution:
$
y\left(\frac{d y}{d x}\right)^2+y\left(\frac{d y}{d x}\right)^4=x
$
Order $=1$, degree $=4$
(vi)
$
x^2 \frac{d^2 y}{d x^2}+\left[1+\left(\frac{d y}{d x}\right)^2\right]^{\frac{1}{2}}=0
$
Solution:
$
x^2 \frac{d^2 y}{d x^2}=-\left[1+\left(\frac{d y}{d x}\right)^2\right]^{1 / 2}
$
Squaring on both sides
$
x^4\left(\frac{d^2 y}{d x^2}\right)^2=\left(1+\left(\frac{d y}{d x}\right)^2\right)
$
Order $=2$, Degree $=2$.

(vii) $\left(\frac{d^2 y}{d x^2}\right)^3=\sqrt{1+\left(\frac{d y}{d x}\right)}$
Solution:
Squaring on both sides
$
\begin{aligned}
& \left(\frac{d^2 y}{d x^2}\right)^6=1+\left(\frac{d y}{d x}\right) \\
& \text { Order }=2 \text {, Degree }=6 \\
& \text { (viii) } \frac{d^2 y}{d x^2}=x y+\cos \left(\frac{d y}{d x}\right) \\
& \text { Solution: } \\
& \text { Order }=2 \\
&
\end{aligned}
$
Since $\frac{d y}{d x}$ cannot be separated from $\cos \left(\frac{d y}{d x}\right)$, degree does not exist.
(ix) $\frac{d^2 y}{d x^2}+5 \frac{d y}{d x}+\int y d x=x^3$
Solution:
Differentiating both sides
$
\begin{aligned}
& \frac{d^3 y}{d x^3}+5 \frac{d^2 y}{d x^2}+d\left(\int y d x\right)=3 x^2 \\
& \text { Order }=3, \text { Degree }=1 .
\end{aligned}
$

(x) $x=e^{x y\left(\frac{d y}{d x}\right)}$
Solution:
$
\begin{aligned}
& \log x=x y\left(\frac{d y}{d x}\right) \\
& \text { Order }=1 \text {, } \\
& \text { degree }=\text { Not exist } \\
&
\end{aligned}
$