Download App

Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}-\text{y}=\text{xe}^{\text{x}}$
✓

Answer

We have, $\frac{\text{dy}}{\text{dx}}-\text{y}=\text{xe}^{\text{x}}\ \dots(1)$ Clearly, it is a linear differential equation of the form $\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$ where $\text{P}=-1$ $\text{Q}=\text{e}^{\text{x}}$ $\therefore$ I.F. $=\text{e}^{\int\text{Pdx}}$ $=\text{e}^{-\int\text{dx}}$ $=\text{e}^{-\text{x}}$Multiplying both sides of (1) by $e^{-x}$, we get
$\text{e}^{-\text{x}}\Big(\frac{\text{dy}}{\text{dx}}-\text{y}\Big)=\text{xe}^{\text{x}}\text{e}^{-\text{x}}$$\Rightarrow\ \text{e}^{-\text{x}}\frac{\text{dy}}{\text{dx}}-\text{e}^{-\text{x}}\text{y}=\text{x}$
Integrating both sides with respect to x, we get
$\text{e}^{-\text{x}}\text{y}=\int\text{xdx + C}$ $\Rightarrow\ \text{e}^{-\text{x}}\text{y}=\frac{\text{x}^2}{2}+\text{C}$$\Rightarrow\ \text{y}=\Big(\frac{\text{x}^2}{2}+\text{C}\Big)\text{e}^{\text{x}}$
Hence, $\text{y}=\Big(\frac{\text{x}^2}{2}+\text{C}\Big)\text{e}^{\text{x}}$ 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 "5 Marks Questions" from Differential Equations→Differential Equations — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

find the area of the region bounded by y = |x - 1| and y = 1.
→
Evaluate the following integrals:
$\int\frac{1}{\text{x}^4+3\text{x}^2+1}\ \text{dx}$
→
A bag contains 3 red and 5 black balls and a second bag contains 6 red and 4 black balls. A ball is drawn from each bag. Find the probability that one is red and the other is black.
→
If A and B are independent events such that P(A) = p, P(B) = 2p and P(Exactly one of A and B occurs) $=\frac{5}{9},$ then find the value or p.
→
Evaluate the following definite integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\sin^3\text{x}\text{ dx}$
→
Find the points o local maxima or local minima, if any, of the following functions, using the first derivatives test. Also, find the local maximum or local minimum values, as the case may be:$f(x) = x^3(x - 1)^2$
→
Evaluate the following integrals:
$\int\limits^{\pi}_0\text{x}\log\sin\text{x}\text{ dx}$
→
If A = $\begin{bmatrix}1&0&2\\0&2&1\\2&0&3\end{bmatrix}$, prove that $A^3 - 6A^2 + 7A + 2I = 0$.
→
A furniture trader deals in only two items – chairs and tables. He has ₹ 50,000 to invest and a space to store at most 35 items. A chair costs him ₹ 1000 and a table costs him ₹ 2000 The trader earns a profit of ₹ 150 and ₹ 250 on a chair and table, respectively. Formulate the above problem as an LPP to maximise the profit and solve it graphically.
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{\sin^{\frac{3}{2}}\text{x}}{\sin^{\frac{3}{2}}\text{x}+\cos^{\frac{3}{2}}\text{x}}\text{ dx}$
→