mathematics-class-12-unit-09-chapter-07-differential-equations-7-8-m-n-3jnrukw

### <Success style="font-size:25px"> Differential Equations L-7 </Success>
### <primary style="font-size:24px"> y'+P(x) y=Q(x) </primary>
<hr>
<main>

- The linear differential equation of the first order is the equation
- $\frac{d y}{d x}+P(x) y=Q(x)$
- where $P(x)$ and $Q(x)$ are functions of $x$ assumed to be at least continuous on an interval $I$. This is an important type of equation and can be solved completely. A close cousin of (3.2) is the Bernoulli equation
- $ d x \mid P(x) y=Q(x) y^n$

</div>
<div style='float: right; width:01%;'>

</main>
<hr>
<footer style = "font-size: 18px"> $\rightarrow$ Differential equation lecture-7 $\rightarrow$ <span style = "color:blue"> y'+P(x) y=Q(x) </span> $\rightarrow$ Integrating factor $\rightarrow$ Remark </footer>

### <Success style="font-size:25px"> Differential Equations L-7 </Success>
### <primary style="font-size:24px"> Integrating factor </primary>
<hr>
<main>

- This suggests that we multiply equation (3.1) by
- $\exp (\phi(x))$
- and equation (3.1) goes over to
- $\text { a } \frac{d}{d x}(y(x) \exp (\phi(x))-Q(x) \exp (\phi(x))$

- If we integrate both sides of (3.5) we get
- $ y(x) \exp (\phi(x))-\int Q(x) \exp (\phi(x)) d x$
- which gives us the solution of the linear differential equation.

</div>
<div style='float: right; width:01%;'>

</main>
<hr>
<footer style = "font-size: 18px">Differential equation lecture-7 $\rightarrow$ y'+P(x) y=Q(x) $\rightarrow$ <span style = "color:blue"> Integrating factor </span> $\rightarrow$ Remark $\rightarrow$ Remark </footer>

### <Success style="font-size:25px"> Differential Equations L-7 </Success>
### <primary style="font-size:24px"> Remark </primary>
<hr>
<main>

- Comments on equation (3.6)

- Since the exponential function does not vanish, equations (3.1) and (3.5) are completely equivalent.

- How are we going to find a $\phi(x)$ such that $\phi^{\prime}(x)=P(x)$ ? Well, if wc cannot, we are simply out of luck! All we can say is take
- $\phi(x)=\int_a^x P(t) d t .$
- and the final solution would have a VERY UGLY APPEARENCE WITH MANY INTEGRATION SIGNS! There is nothing one can do except to deal with it!

</div>
<div style='float: right; width:01%;'>

</main>
<hr>
<footer style = "font-size: 18px">y'+P(x) y=Q(x) $\rightarrow$ Integrating factor $\rightarrow$ <span style = "color:blue"> Remark </span> $\rightarrow$ Remark $\rightarrow$ Summary </footer>

### <Success style="font-size:25px"> Differential Equations L-7 </Success>
### <primary style="font-size:24px"> Remark </primary>
<hr>
<main>

- <ins>	Comments contd</ins>

- We shall assume that we can find such a $\phi(x)$ namely $\int P(x) d x$ and and rewrite the solution obtained as
- $\left.y(x) \exp \left(\int P(x) d x\right)\right)-\int Q(x) \exp \left(\int P(x) d x\right) d x$

- It is understood that in both occurrences of $\int P(x) d x$ are identical and so the SAME constant of integration should be assigned for both

</div>
<div style='float: right; width:01%;'>

</main>
<hr>
<footer style = "font-size: 18px">Integrating factor $\rightarrow$ Remark $\rightarrow$ <span style = "color:blue"> Remark </span> $\rightarrow$ Summary $\rightarrow$ Examples </footer>

### <Success style="font-size:25px"> Differential Equations L-7 </Success>
### <primary style="font-size:24px"> Summary </primary>
<hr>
<main>

- <ins>Comments contd:</ins>

- Here we strongly advise the student not to memorize the final formula but rather, follow the steps:
- First compute $f P(x) d x$ ignoring the constant of integration
- Multiply the differential equation by $\exp \left(\int P(x) d x\right)$.
- Perform the final integration - and in case there are initial conditions given, employ definite intgrals.

</div>
<div style='float: right; width:01%;'>

</main>
<hr>
<footer style = "font-size: 18px">Remark $\rightarrow$ Remark $\rightarrow$ <span style = "color:blue"> Summary </span> $\rightarrow$ Examples $\rightarrow$ Examples </footer>

### <Success style="font-size:25px"> Differential Equations L-7 </Success>
### <primary style="font-size:24px"> Examples </primary>
<hr>
<main>

- Example 1: Solve the equation on the interval $(-\pi / 2, \pi / 2)$ :
- $\frac{d y}{d x}+(\tan x) y=\sin x$

- Solution: Here $P(x)-\tan x$ and so
- $\exp \int P(x) d x=\exp (\log (\sec x))=\sec x \text {. }$

- Ignored the const. of integrationl Multiply the diff. eqn. by sec $x$ to get
- $\frac{d}{d x}(y \sec x)=\sin x \sec x=\tan x$

- Integratang (constant of integration is needed nowl) and smplifying we get
- $ y(x)-\cos x(\log (\sec x)+c)$

</div>
<div style='float: right; width:01%;'>

</main>
<hr>
<footer style = "font-size: 18px">Remark $\rightarrow$ Summary $\rightarrow$ <span style = "color:blue"> Examples </span> $\rightarrow$ Examples $\rightarrow$ Examples </footer>

### <Success style="font-size:25px"> Differential Equations L-7 </Success>
### <primary style="font-size:24px"> Examples </primary>
<hr>
<main>

- Example 2: Note that the problem has been modilied here. Given that $y(x)$ is a continuous real valued function defined on $(0, \infty)$ such that
- $ 6 \int_1^x y(t) d t=3 x y(x)-x^3, \quad x>0,$
- calculate the value $y(2)$.
- Solution: Note that the integral is meaningful since $y(x)$ is given to be continuous. Also note that for any continuous function $g$ defined on $(0, \infty)$, the integral
- $\int_1^x g(t) d t$
- is differentiable and its derivative is $g(x)$. Thus the left hand side of $(3.10)$ is differentiable and hence $3 x y(x)$ is differentiable and so is $y(x)$ on $(0, \infty)$. Differentiating $(3.10)$ produces a linear differential equation.

</div>

</main>
<hr>
<footer style = "font-size: 18px">Summary $\rightarrow$ Examples $\rightarrow$ <span style = "color:blue"> Examples </span> $\rightarrow$ Examples $\rightarrow$ Examples </footer>

### <Success style="font-size:25px"> Differential Equations L-7 </Success>
### <primary style="font-size:24px"> Examples </primary>
<hr>
<main>

- <ins>Example from JEE 2014 (paper - 2)</ins>

- Given that $f(x)$ is the solution of the differential equation and the given initial condition on $(-1,1)$.
- $\frac{d y}{d x}+\frac{x y}{x^2-1}=\frac{x^4+2 x}{\sqrt{1-x^2}}, y(0)-0 .$

- Find the value of the integral
- $\int_p^y f(x) d x$

- The given equation is a linear differential equation which is defined for $1<x<1$. Here wo compute
- $\int P(x) d x-\frac{1}{2} \int_1^{-2 } x^2-\frac{1}{2} \log \left(1 \quad x^2\right) x d x$

</div>
<div style='float: right; width:01%;'>

</main>
<hr>
<footer style = "font-size: 18px">Examples $\rightarrow$ Examples $\rightarrow$ <span style = "color:blue"> Examples </span> $\rightarrow$ Examples $\rightarrow$ Examples </footer>

### <Success style="font-size:25px"> Differential Equations L-7 </Success>
### <primary style="font-size:24px"> Examples </primary>
<hr>
<main>

- <ins>Example from JEE 2014 (paper - 2)</ins>

- Given that $f(x)$ is the solution of the differential equation and the given initial condition on $(-1,1)$.
- $\frac{d y}{d x}+\frac{x y}{x^2-1}=\frac{x^4+2 x}{\sqrt{1-x^2}}, y(0)-0 .$

- Find the value of the integral
- $\int_p^y f(x) d x$

- The given equation is a linear differential equation which is defined for $1<x<1$. Here wo compute
- $\int P(x) d x-\frac{1}{2} \iint_1^{-2 x d x} x^2-\frac{1}{2} \log \left(1 \quad x^2\right) .$

</div>
<div style='float: right; width:01%;'>

</main>
<hr>
<footer style = "font-size: 18px">Examples $\rightarrow$ Examples $\rightarrow$ <span style = "color:blue"> Examples </span> $\rightarrow$ Examples $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Differential Equations L-7 </Success>
### <primary style="font-size:24px"> Examples </primary>
<hr>
<main>

- <ins>Example from JEE 2014 (paper - 2) contd</ins>

- Hence we should multiply the differential equation by the factor
- $\exp \int P(x) d x=\sqrt{1-x^2}$

- After multiplying we see that
- $\frac{d}{d x}\left(y \sqrt{1-x^2}\right)-x^4+2 x$

- We now integrate this from 0 to $x$ and we get
- $- f(x) \sqrt{1-x^2}=f(0)+\int_0^x\left(t^4+2 t\right) d t=\frac{x^5}{5}+x^2 .$

- Now what is asked is the integral of the function $f(x)$ over an interval symmetric about the origin and so the odd part of $f(x)$ namely the term coming from $x^5 / 5$ would be zero.

</div>
<div style='float: right; width:01%;'>

</main>
<hr>
<footer style = "font-size: 18px">Examples $\rightarrow$ Examples $\rightarrow$ <span style = "color:blue"> Examples </span> $\rightarrow$ Example $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Differential Equations L-7 </Success>
### <primary style="font-size:24px"> Example </primary>
<hr>
<main>

- <ins>Example JFE 2016, paper 1</ins>

- With some notational changes, the linear differential equation on $(0, \infty)$ is
- $ y^{\prime}(x)=2-\frac{y(x)}{x}$
- (i) Calculate the limits as $x \rightarrow 0+$ of $y^{\prime}(1 / x), x y(1 / x)$ and $x^2 y^{\prime}(x)$
- (ii) Is the function $|y(x)|$ bounded on $(0.2)$ ?

- Solution: Rewriting the equation as
- $ y^{\prime}(x)+\frac{y(x)}{x}-2$

- We see that $\exp \left(\int P(x) d x\right)=x$ Multiplying the equation by $x$ we get
- $ d x(x y(x))=2 x$

- As such, $y(1)$ can be assigned any real value. Let us assume that $y(1)=a$

</div>
<div style='float: right; width:01%;'>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Examples $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example JEE 2016, Paper - I contd... $\rightarrow$ The Bernoulli Equation </footer>

### <Success style="font-size:25px"> Differential Equations L-7 </Success>
### <primary style="font-size:24px"> The Bernoulli Equation </primary>
<hr>
<main>

- This is the equation
- $\frac{d y}{d x}+P(x) y=Q(x) y^n$
- where $P(x)$ and $Q(x)$ are continuous on an interval $I$.
- If $n=0$ or $n=1$ then the equation is linear and so we shall assume $n \neq 0,1$. Dividing by $y^n$ we get
- $\frac{1}{y^n} \frac{d y}{d x}+P(x) y^{1-n}=Q(x)$

- The substitution $u=y^{1-n}$ converts equation (3.15) into a linear differential equation.
- Warning: If $n>0$ we are tacitly assuming that $y(x) \neq 0$ Suppose we are given initial conditions such as $y\left(x_0\right)=0$ then the method cannot be employed !
</div>

</main>
<hr>
<footer style = "font-size: 18px">Example $\rightarrow$ Example JEE 2016, Paper - I contd... $\rightarrow$ <span style = "color:blue"> The Bernoulli Equation </span> $\rightarrow$ Reduction to a linear equation $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Differential Equations L-7 </Success>
### <primary style="font-size:24px"> Example </primary>
<hr>
<main>

- (4) Solve the equation
- $\frac{d y}{d t}=y(1-y), \quad y(0) \neq 0$
- (i) By the method of separation of variables. Left for the student.
- (ii) As a Bernoulli equation.

- Solution (ii): The condition $y(0) \neq 0$ allows us to divide by $y^2$. The equation can be written as
- $\frac{-y^{\prime}}{y^2}+\frac{1}{y}=1$

- Put $1 / y=u$ and we get the equation $u^{\prime}+u=1$ whose solution is $u=1+c \exp (-x)$ and so we get
- $ y(x)=\frac{1}{11 \operatorname{cexp}(x)}$
</div>
<div style='float: right; width:01%;'>

</main>
<hr>
<footer style = "font-size: 18px">The Bernoulli Equation $\rightarrow$ Reduction to a linear equation $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Exercises $\rightarrow$ Exercises </footer>

### <Success style="font-size:25px"> Differential Equations L-7 </Success>
### <primary style="font-size:24px"> Exercises </primary>
<hr>
<main>

- <ins>Discussion of Exercise (6)</ins>
- $\frac{d y}{d x}+2 x \tan y=(\sec y) e^{x^2}$

- Multiplying the Diff. Eqn. by $\cos y$ gives
- $ (\cos y) \frac{d y}{d x}+2 x \sin y=e^{-x^2}$

- Let us now put $\sin y=u$. Then
- $\cos y \frac{d y}{d x}-d x$

- The diff eqn transforms into the linear equation
- $\frac{d u}{d x}+2 x u-e^{-x^2}$

</div>
<div style='float: right; width:01%;'>

</main>
<hr>
<footer style = "font-size: 18px">Example $\rightarrow$ Exercises $\rightarrow$ <span style = "color:blue"> Exercises </span> $\rightarrow$ Thank you $\rightarrow$  </footer>