Find the equation of a curve passing through | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The given differential equation is $y^{\prime} = e^{x} \sin x$.This can be written as $\frac{dy}{dx} = e^{x} \sin x$.Integrating both sides with r

Vedclass · Accepted Answer

$2y = e^{x}(\sin x - \cos x) + 1$

Vedclass · Answer

(A) The given differential equation is $y^{\prime} = e^{x} \sin x$.This can be written as $\frac{dy}{dx} = e^{x} \sin x$.Integrating both sides with respect to $x$:$y = \int e^{x} \sin x \, dx + C$  ...........$(1)$To evaluate $I = \int e^{x} \sin x \, dx$,we use integration by parts:$I = \sin x \cdot e^{x} - \int \cos x \cdot e^{x} \, dx$$I = e^{x} \sin x - [\cos x \cdot e^{x} - \int(-\sin x) \cdot e^{x} \, dx]$$I = e^{x} \sin x - e^{x} \cos x - I$$2I = e^{x}(\sin x - \cos x)$$I = \frac{e^{x}(\sin x - \cos x)}{2}$Substituting this into equation $(1)$:$y = \frac{e^{x}(\sin x - \cos x)}{2} + C$  ...........$(2)$Since the curve passes through $(0,0)$,we substitute $x=0$ and $y=0$:$0 = \frac{e^{0}(\sin 0 - \cos 0)}{2} + C$$0 = \frac{1(0 - 1)}{2} + C$$0 = -\frac{1}{2} + C \Rightarrow C = \frac{1}{2}$Substituting $C = \frac{1}{2}$ into equation $(2)$:$y = \frac{e^{x}(\sin x - \cos x)}{2} + \frac{1}{2}$$2y = e^{x}(\sin x - \cos x) + 1$$2y - 1 = e^{x}(\sin x - \cos x)$.

Find the equation of a curve passing through the point $(0,0)$ and whose differential equation is $y^{\prime}=e^{x} \sin x$.

Similar Questions

The rate of disintegration of a radioactive element at time $t$ is proportional to its mass at that time. Then the time during which the original mass of $1.5 \text{ g}$ will disintegrate into its mass of $0.5 \text{ g}$ is proportional to

If the population grows at the rate $5 \%$ per year,then the time taken for the population to become double is $\quad$ (Given $\log 2 = 0.6912$) (in $years$)

The rate of disintegration of a radioactive element at any time $t$ is proportional to its mass $M$ at that time. If $k (k > 0)$ is the constant of proportionality, find the time required for an initial mass of $1.5 \text{ g}$ to disintegrate to a mass of $0.5 \text{ g}$.

$A$ curve is such that the area of the region bounded by the coordinate axes,the curve,and the ordinate of any point on it is equal to the cube of that ordinate. The curve represents:

If $\frac{d^{2} y}{d x^{2}}=\sin x+e^{x}$,$y(0)=3$,and $\frac{d y}{d x}$ at $x=0$ is $4$,then find the equation of the curve.

Vedclass Test Series

Exam Paper Generator

Online Exam Module