Identify the statement(s) which is/are true?

Important Questions on Differential Equations

HARD

Mathematics

IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 25 - Differential Equations>Exercise 1>Q 16

Let $f:\left(0,\infty \right)\to R$ be a differentiable function such that $f^{\text{'}}\left(x\right)=2-\frac{f\left(x\right)}{x}$ for all $x\in \left(0,\infty \right)$ and $f\left(1\right)\ne 1.$Then

HARD

Mathematics

IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 25 - Differential Equations>Exercise 1>Q 17

Let $f : [0,\infty )\to R$ be a continuous function such that $f\left(x\right)=1-2x+{\int }_0^x{e}^{x-t} f\left(t\right) dt$ for all $x\in [0,\infty ).$ Then, which of the following statement(s) is (are) TRUE?

HARD

Mathematics

IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 25 - Differential Equations>Exercise 1>Q 18

A differentiable function satisfies $f\left(x\right)={\int }_0^x(f\left(t\right)\cos t-\cos \left(t-x\right))dt$. Which of the following hold(s) good?

HARD

Mathematics

IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 25 - Differential Equations>Exercise 1>Q 19

$y=f\left(x\right)$ has differential equation $y\left(2xy+e^{\mathrm{x}}\right)dx-e^{\mathit{x}}dy=0$, which is passing through the point $(0, 1).$ Then which of the following is/are true about the function?

HARD

Mathematics

IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 25 - Differential Equations>Exercise 1>Q 20

A solution curve of the differential equation $\left(x^2+xy+4x+2y+4\right)\frac{dy}{dx}-y^2=0, x>0$, passes through the point $\left(1,3\right)$. Then the solution curve

HARD

Mathematics

IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 25 - Differential Equations>Exercise 1>Q 21

Let $f:\left(0, \pi \right)\to \mathbb{R}$ be a twice differentiable function such that $\underset{t\to x}{\lim }\left\{\frac{f\left(x\right)\sin t-f\left(t\right)\sin x}{t-x}\right\}={\sin }^{2}x$ for all $x\in \left(0,\pi \right)$. If $f\left(\frac{\pi }{6}\right)=-\frac{\pi }{12}$, then which of the following statement(s) is (are) TRUE ?

HARD

Mathematics

IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 25 - Differential Equations>Exercise 1>Q 22

Consider the family of all circles whose centres lie on the straight line $y=x.$ If this family of circles is represented by the differential equation $P{y}^{"}+Q{y}^{\text{'}}+1=0,$ where $P, Q$ are the functions of $x, y$ and $y^{\text{'}}$ (here, $y^{\text{'}}=\frac{dy}{dx}, y^{"}=\frac{d^{2}y}{d{x}^2}$) then which of the following statement(s) is/are true?

HARD

Mathematics

IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 25 - Differential Equations>Exercise 1>Q 23

A tangent drawn to the curve $y=f\left(x\right)$ at $P(x,y)$ cuts the $x$-axis and $y$-axis at $A$ and $B$, respectively, such that $BP:AP=3:1$. Given that $f\left(1\right)=1$, then