Linear Differential Equations

For $x\in \mathrm{ℝ}$, let $y\left(x\right)$ be a solution of the differential equation $\left(x^2-5\right)\frac{dy}{dx}-2xy=-2x{\left(x^2-5\right)}^2$ such that $y\left(2\right)=7$. Then the maximum value of the function $y\left(x\right)$ is

Let $f:[1.\infty )\to \mathrm{ℝ}$ be a differentiable function such that $f\left(1\right)=\frac{1}{3}$ and $3{\int }_1^{x}f\left(t\right)dt=xf\left(x\right)-\frac{x^3}{3},x\in [1,\infty )$. Let $e$ denote the base of the natural logarithm. Then the value of $f\left(e\right)$ is

Solve the differential equation $\frac{dy}{dx}-\frac{3y}{x}+\frac{6}{x}=0$.

Solve the differential equation $\frac{dy}{dx}=-\left[\frac{ x+y\cos x}{1+\sin x}. \right]$

Solve: $\frac{dy}{dx}+\left({\sec }^{2}x\right)y=\sec x\tan x$.

Consider $g\left(x\right)=\left\{\begin{array}{l}\sin x; 0\le x<\frac{\mathrm{\pi }}{2}\\ \cos x; x\ge \frac{\mathrm{\pi }}{2}\end{array}\right.$ and a continuous function $y=f\left(x\right)$ satisfies $5\frac{dy}{dx}+5y=g\left(x\right)$; $f\left(0\right)=0$, then

The solution of $\frac{dy}{dx}=\frac{x^2-y^2+1}{2xy}$

Let the solution curve $y=y\left(x\right)$ of the diffrential equation $\left(1+e^{2x}\right)\left(\frac{dy}{dx}+y\right)=1$ pass through the point $\left(0,\frac{\pi }{2}\right)$. Then $\underset{x\to \infty }{\lim }{e}^{x}y\left(x\right)$ is equal to:

For $x\in \mathrm{ℝ}$, let the function $y\left(x\right)$ be the solution of the differential equation $\frac{dy}{dx}+12y=\cos \left(\frac{\pi }{12}x\right),y\left(0\right)=0.$ Then, which of the following statements is/are TRUE?

Let $y=y\left(x\right), x>1$, be the solution of the differential equation $\left(x-1\right)\frac{dy}{dx}+2xy=\frac{1}{x-1}$, with $y\left(2\right)=\frac{1+e^4}{2e^4}$. If $y\left(3\right)=\frac{e^{\alpha }+1}{\beta e^{\alpha }}$, then the value of $\alpha +\beta$ is equal to

Consider the solutions set $f\left(x,y\right)=0$ of the differential equation $x^{2}dx-y^{2}dx-x\left(4dy-2ydy\right)+4\left(ydx-dx\right)=0$ then which of the following is/are true?

If a continuous function $f\left(x\right)$ satisfies the relation ${\int }_0^{x}tf\left(x-t\right)dt={\int }_0^{x}f\left(t\right)dt+\sin x+\cos x-x-1$ for all real values of $x$ then which of the following doesn't hold good

Let $y=f\left(x\right)$ is a curve such that slope of tangent at any point is equal to sum of ratio of ordinate and abscissa of that point and square of its abscissa. If the curve passes through $\left(1,-2\right)$, then $2{\int }_0^{1}f\left(x\right)dx$ is $\lambda$. Find $\left|\lambda \right|$

Let $f:R\to R$ be defined by $f\left(x\right)=e^{x-1}-e^{-\left|x-1\right|}$ and $g:R\to R$ is a differentiable function such that $g\left(1\right)=1$ and satisfy differential equation $2e^{x+1}\frac{dy}{dx}+e^2-e^{2x}=0$, then area of region in the first quadrant bounded by the curves $y=f\left(x\right), y=g\left(x\right)$ and $x=0$ is ___________

If the integrating factor of $x\left(1-x^2\right)dy+\left(2x^{2}y-y-a{x}^3\right)dx=0$ is $e^{\int P·dx}$, then $P$ is equal to

Solution of the differential equation $\left(1+y^2\right)dx=\left({\tan }^{-1}y-x\right)dy$ is
(where $c$ is an arbitrary constant)

Consider the differential equation, $ydx-\left(x+y^2\right)dy=0$. If for $y=1, x$ takes value $1$, then value of $x$ when $y=4$ is

Let $y=y\left(x\right)$ be the solution of the differential equation $x{\log }_{e}x\frac{dy}{dx}+y=3x{\log }_{e}x, \left(x>1\right).$ If $y\left(e\right)=0,$ then $y\left(e^2\right)$ is equal to

The integrating factor (I.F.) of differential equation $\frac{dy}{dx}(1+x)-xy=1-x$ is $\dots \dots ..$

If $f\left(x\right)$ is the solution of the equation $\left(x+1\right)\frac{dy}{dx}-xy-1=0$ and $f\left(0\right)=0$, then $f\left(e\right)$ equals