Solve the following differential equation (x+2y^2) dy dx =y, give… - Vidyadip

Question

Solve the following differential equation $(\text{x}+2\text{y}^2)\frac{\text{dy}}{\text{dx}}=\text{y},$ given that when x = 2, y = 1.

✓

Answer

We have, $(\text{x}+2\text{y}^2)\frac{\text{dy}}{\text{dx}}=\text{y}$ $\Rightarrow\ \frac{\text{dx}}{\text{dy}}=\frac{1}{\text{y}}(\text{x}+2\text{y}^2)$ $\Rightarrow\ \frac{\text{dx}}{\text{dy}}-\frac{1}{\text{y}}\text{x}=2\text{y}\ \dots(1)$ Clearly, it is a linear differential equation of the form $\frac{\text{dx}}{\text{dy}}+\text{Px = Q}$ where $\text{P}=-\frac{1}{\text{y}}$ $\text{Q}=2\text{y}$ $\therefore$ I.F. $=\text{e}^{\int\text{Pdy}}$ $=\text{e}^{-\int\frac{1}{\text{y}}\text{dy}}$ $=\text{e}^{-\log\text{y}}$ $=\frac{1}{\text{y}}$Multiplying both sides of (1) by $\frac{1}{\text{y}},$ we get
$\frac{1}{\text{y}}\Big(\frac{\text{dx}}{\text{dy}}-\frac{1}{\text{y}}\text{x}\Big)=\frac{1}{\text{y}}\times2\text{y}$ $\Rightarrow\ \frac{1}{\text{y}}\frac{\text{dx}}{\text{dy}}-\frac{1}{\text{y}^2}\text{x}=2$ Integrating both sides with respect to y, we get $\text{x}\frac{1}{\text{y}}=\int2\text{dy + C}$ $\Rightarrow\ \text{x}\frac{1}{\text{y}}=2\text{y + C}$ $\Rightarrow\ \text{x}=2\text{y}^2+\text{Cy}\ \dots(2)$ Now, $\text{y}=1$ at $\text{x}=2$ $\therefore\ 2=2+\text{C}$ $\Rightarrow\ \text{C}=0$ Putting the value of C in (2), we get $\text{x}=2\text{y}^2$ Hence, $\text{x}=2\text{y}^2$ is the required solution.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(5 Marks)" from Differential Equations→Differential Equations — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Show that $\text{f(x)}=\cos\text{x}^2$ is a continuous function.

A firm manufactures two products A and B on which profit earned per unit ₹3/- and ₹4/- respectively. Each product is processed on two machines M1 and M2. The product A requires one minute of processing time on M1 and two minute of processing time on M2, B requires one minute of processing time on M1 and one minute of processing time on M2. Machine M1 is available for use for 450 minutes while M2 is available for 600 minutes during any working day. Find the number of units of product A and B to be manufactured to get the maximum profit.

By computing the shortest distance determine whether the following pairs of lines intersect or not:
$\frac{\text{x}-1}{2}=\frac{\text{y}+1}{3}=\text{z}$ and $\frac{\text{x}+1}{5}=\frac{\text{y}-2}{1};\text{z}=2$

An item is manufactured by three machines $\mathrm{A}, \mathrm{B}$ and $C$ . Out of the total number of items manufactured during a specified period, $50 \%$ are manufactured on machine A, $30 \%$ on B and $20 \%$ on $C$, $2 \%$ of the items produced on A and $2 \%$ of items produced on $B$ are defective and $3 \%$ of these produced on $C$ are defective. All the items stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine $A$ ?

Find the equation of the plane through the points $(2,2,-1)$ and $(3,4,2)$ and parallel to the line whose direction ratios are $7,0,6$.

Evaluate the following integrals:$\int^\limits{\frac{1}{2}}_{0}\frac{1}{(1+\text{x}^2)\sqrt{1-\text{x}^2}}\text{ dx}$

Solve the following differential equations:$\tan\text{y}\frac{\text{dy}}{\text{dx}}=\sin(\text{x}+\text{y})+\sin(\text{x}-\text{y})$

Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix}$

For the following differntial equations verify that the accompanying function is a solution:

Differential equation	Function
$\text{y}=\Big(\frac{\text{dy}}{\text{dx}}\Big)^2$	$\text{y}=\frac{1}{4}(\text{x}\pm\text{a})^2$

Show that the three lines with direction cosines $\frac{12}{13},\frac{-3}{13},\frac{-4}{13},\frac{4}{13},\frac{12}{13},\frac{3}{13},\frac{3}{13},\frac{-4}{13},\frac{12}{13}$ are mutually perpendicular.