SATHEE: Differential Equations 2 Question 24

Differential Equations 2 Question 24

25. Let $y^{'} (x) + y (x) g^{'} (x) = g (x) g^{'} (x), y (0) = 0, x \in R$ , where $f^{'} (x)$ denotes $\frac{d f (x)}{d x}$ and $g (x)$ is a given non-constant differentiable function on $R$ with $g (0) = g (2) = 0$ . Then, the value of $y (2)$ is ……

(2011)

Show Answer

Solution:

$\frac{d y}{d x} + y \cdot g^{'} (x) = g (x) g^{'} (x)$

$I F = e^{\int g^{'} (x) d x} = e^{g (x)}$

$∴$ Solution is $y (e^{g (x)}) = \int g (x) \cdot g^{'} (x) \cdot e^{g (x)} d x + C$

$\begin{aligned} Put \begin{aligned} g (x) & = t, g^{'} (x) d x = d t \\ y (e^{g (x)}) & = \int t \cdot e^{t} d t + C \\ = t \cdot e^{t} - \int 1 \cdot e^{t} d t + C & = t \cdot e^{t} - e^{t} + C \\ y e^{g (x)} & = (g (x) - 1) e^{g (x)} + C \\ y (0) & = 0, g (0) = g (2) = 0 \end{aligned} \end{aligned}$

$∴$ Eq. (i) becomes,

$\begin{aligned} y (0) \cdot e^{g (0)} & = (g (0) - 1) \cdot e^{g (0)} + C \\ \Rightarrow & 0 & = (- 1) \cdot 1 + C \Rightarrow C = 1 \\ ∴ & y (x) \cdot e^{g (x)} & = (g (x) - 1) e^{g (x)} + 1 \\ \Rightarrow & y (2) \cdot e^{g (2)} & = (g (2) - 1) e^{g (2)} + 1, where g (2) = 0 \\ \Rightarrow & y (2) \cdot 1 & = (- 1) \cdot 1 + 1 \\ y (2) & = 0 \end{aligned}$

Differential Equations 2 Question 24

25. Let $y^{'} (x) + y (x) g^{'} (x) = g (x) g^{'} (x), y (0) = 0, x \in R$ , where $f^{'} (x)$ denotes $\frac{d f (x)}{d x}$ and $g (x)$ is a given non-constant differentiable function on $R$ with $g (0) = g (2) = 0$ . Then, the value of $y (2)$ is ……

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Differential Equations 2 Question 24

25. Let y′(x)+y(x)g′(x)=g(x)g′(x),y(0)=0,x∈R, where f′(x) denotes df(x)dx and g(x) is a given non-constant differentiable function on R with g(0)=g(2)=0. Then, the value of y(2) is ……

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

25. Let $y^{'} (x) + y (x) g^{'} (x) = g (x) g^{'} (x), y (0) = 0, x \in R$ , where $f^{'} (x)$ denotes $\frac{d f (x)}{d x}$ and $g (x)$ is a given non-constant differentiable function on $R$ with $g (0) = g (2) = 0$ . Then, the value of $y (2)$ is ……