Question
Solve the following differential equation:
$(\text{x}+\text{y})^2\frac{\text{dy}}{\text{dx}} = 1$
$(\text{x}+\text{y})^2\frac{\text{dy}}{\text{dx}} = 1$
✓
Answer
$(\text{x}+\text{y})^2\frac{\text{dy}}{\text{dx}} = 1$
Let $\text{x}+\text{y} = \text{v}$
$1 + \frac{\text{dy}}{\text{dx}} = \frac{\text{dv}}{\text{dx}}$
$\frac{\text{dy}}{\text{dx}} = \frac{\text{dv}}{\text{dx}} - 1$
So,
$\text{v}^2\Big(\frac{\text{dv}}{\text{dx}}-1\Big) = 1$
$\frac{\text{dv}}{\text{dx}} = \frac{1}{\text{v}^2}+1$
$\frac{\text{dv}}{\text{dx}} = \frac{\text{v}^2+1}{\text{v}^2}$
$\frac{\text{v}^2}{\text{v}^2+1}\text{dv} = \text{dx}$
$\int\frac{\text{v}^2+1-1}{\text{v}^2+1}\text{dv} = \int \text{dx}$
$\int\Big(1-\frac{1}{\text{v}^2+1}\Big)\text{dv} = \int\text{dx}$
$\text{v}-\tan^{-1}(\text{v}) = \text{x} + \text{C}$
$\text{x}+\text{y}-\tan^{-1}(\text{x}+\text{y}) = \text{x}+\text{C}$
$\text{y}-\tan^{-1}(\text{x}+\text{y}) = \text{C}$
Let $\text{x}+\text{y} = \text{v}$
$1 + \frac{\text{dy}}{\text{dx}} = \frac{\text{dv}}{\text{dx}}$
$\frac{\text{dy}}{\text{dx}} = \frac{\text{dv}}{\text{dx}} - 1$
So,
$\text{v}^2\Big(\frac{\text{dv}}{\text{dx}}-1\Big) = 1$
$\frac{\text{dv}}{\text{dx}} = \frac{1}{\text{v}^2}+1$
$\frac{\text{dv}}{\text{dx}} = \frac{\text{v}^2+1}{\text{v}^2}$
$\frac{\text{v}^2}{\text{v}^2+1}\text{dv} = \text{dx}$
$\int\frac{\text{v}^2+1-1}{\text{v}^2+1}\text{dv} = \int \text{dx}$
$\int\Big(1-\frac{1}{\text{v}^2+1}\Big)\text{dv} = \int\text{dx}$
$\text{v}-\tan^{-1}(\text{v}) = \text{x} + \text{C}$
$\text{x}+\text{y}-\tan^{-1}(\text{x}+\text{y}) = \text{x}+\text{C}$
$\text{y}-\tan^{-1}(\text{x}+\text{y}) = \text{C}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Using differentials, find the approximate values of the following:
$\log_\text{e}4.04,$ it being given that $\log_{{10}^{4}}=0.6021$ and $\log_{10}\text{e}=0.4343$
→$\log_\text{e}4.04,$ it being given that $\log_{{10}^{4}}=0.6021$ and $\log_{10}\text{e}=0.4343$
Evaluate the following integrals:
$\int\limits_{0}^{1}\tan^{-1}\text{x dx}$
→$\int\limits_{0}^{1}\tan^{-1}\text{x dx}$
If $\big|\vec{\text{a}}+\vec{\text{b}}\big|=60,\big|\vec{\text{a}}-\vec{\text{b}}\big|=40$ and $\big|\vec{\text{b}}\big|=46,$ find $|\vec{\text{a}}|$
→Two bad eggs are accidently mixed up with ten good ones. Three eggs are drawn at random with replacement from this lot. Compute the mean for the number of bad eggs drawn.
A manufacturer produces three types of bolts, $x, y$ and $z$ which he sells in two markets. Annual sales (in ₹) are indicated below:
If unit sales prices of $x, y$ and z are ₹ 2.50, ₹ $1.50$ and ₹ $1.00$ respectively, then answer the following questions using the concept of matrices.
→
| Markets | Products | ||
| $X$ | $Y$ | $Z$ | |
| I | $10000$ | $2000$ | $18000$ |
| II | $6000$ | $20000$ | $8000$ |
- Find the total revenue collected from the Market-I.
- $₹\ 44000$
- $₹\ 48000$
- $₹\ 46000$
- $₹\ 53000$
- Find the total revenue collected from the Market-II.
- $₹\ 51000$
- $₹\ 53000$
- $₹\ 46000$
- $₹\ 49000$
- If the unit costs of the above three commodities are $₹\ 2.00, ₹\ 1.00$ and $50$ paise respectively, then find the gross profit from both the markets.
- $₹\ 53000$
- $₹\ 46000$
- $₹\ 34000$
- $₹\ 32000$
- If matrix $\text{A}=[\text{a}_\text{ij}]_{2\times2},$ where $\text{a}_\text{ij}=1,$ if $\text{i}\neq\text{j}$ and $\text{a}_\text{ij}=0,$ if $\text{i}=\text{j}$ then $A^2$ is equal to:
- $I$
- $A$
- $OR$
- None of these
- If $A$ and $B$ are matrices of same order, then $(AB' - BA')$ is a.
- Skew-synunetric matrix.
- Null matrix.
- Symmetric matrix.
- Unit matrix.
A company produces two types of goods, A and B, that require gold and silver. Each unit of type A requires 3gm of silver and 1 gm of gold while that of type B requires 1 gm of silver and 2gm of gold. The company can produce 9gm of silver and 8gm of gold. If each unit of type A brings a profit of Rs. 40 and that of type B Rs. 50, find the number of units of each type that the company should produce to maximize the profit. What is the maximum profit?
Evaluate the following integrals:$\int\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)\text{dx}$
→Mother, father and son line up at random for a family picture. If A and B are two events given by A = Son on one end, B = Father in the middle, find P (A/B) and P (B/A).
Solve the following initial value problems:
$\text{y}'+\text{y}=\text{e}^{\text{x}},\text{ y}(0)=\frac{1}{2}$
→$\text{y}'+\text{y}=\text{e}^{\text{x}},\text{ y}(0)=\frac{1}{2}$
Find the equatoion of the plane passing through the points $(2, 2, 1)$ and $(9, 3, 6)$ and perpendicular to the plane $2x + 6y + 6z = 1.$
→