We will cover following topics

Formulation Of Differential Equations

A differential equation (DE) is a relation that contains a finite set of functions and their derivatives with respect to one or more independent variables.

An ordinary differential equation (ODE) is a relation that contains derivatives of functions of only one variable.

The highest derivative that appear in a ODE is the order of that ODE.

If the ODE can be written as $a_{0}(x) y^{(n)}(x)$+$a_{1}(x) y^{(n-1)}(x)$+$\cdots$+$a_{n-1}(x) y^{(1)}(x)$+$a_{n}(x) y(x)$=$F(x)$, then the given ODE is linear.

If such representation is not possible, then we say that the given ODE is nonlinear.

Equations Of First Order And First Degree

Method of Variation of Parameters: Consider the equation $\dfrac{dy}{dx}+P(x)y=Q(x)$.

The homogeneous solution is calculated by putting $Q(x)=0$ in the given differential equation.

Solving for $y$, we get $y=A e^{P(x)}$.

Now, according to the method of variation of parameters, we write the solution as:

$y=v(x)e^{P(x)}$ where $A$ is replaced by $v(x)$.

The complete solution is written as:

\[y= (v(x)+A) e^{- \int P(x)}\]

where $v(x)=e^{P(x)}$, $v'(x)=\dfrac{Q(x)}{e^{- \int P(x)} }$

Separable Equations: These are equations of the forms $y^{\prime}=f(x) g(y)$ and can be solved as $\int \dfrac{d y}{g(y)}=\int f(x) d x+C$, provided $g(y) \neq 0$.

If an ODE is of the form $y^{\prime}=F(a x+b y+c)$, it can be reduced to the separable form by substituting $ax+by+c=v$.

Similarly, if the ODE is of the form $y^{\prime}=f(y / x)$, it can be reduced to the separable form by substituting $v=y/x$.

Exact Equation: A first order ODE of the form $M(x, y) d x+N(x, y) d y=0$ is exact if there exists a function $u(x,y)$ such that

$M=\dfrac{\partial u}{\partial x}$ and $N=\dfrac{\partial u}{\partial y}$, so that the given ODE can be written as $du=0$ and the solution becomes $u=c$.

Theorem: Let $M$ and $N$ be defined and continuously differentiable on a rectangle $R=\left\{(x, y) :\vert x-x_{0} \vert < a , \vert y-y_{0} \vert < b\right\}$.

Then $Mdx+Ndy=0$ is exact if and only if:

\[\partial M / \partial y=\partial N / \partial x \text{ } \forall (x,y) \in R\]

Integrating Factor

Integrating Factor*: An integrating factor is a function by which an ordinary differential equation can be multiplied in order to make it integrable

Exact Differential Equation: The differential equation $Mdx+Ndy=0$ is said to be exact if $M_y=N_x$.

If the equation is not exact, it can be made exact by multiplying the entire equation by $\mu (x,y)$ such that the resultant equation $(\mu M) d x+(\mu N) d y=0$. Here, $\mu$ is called the integrating factor.

Two special cases arise when the given equation is not exact, i.e. $M_y \neq N_x$:

Case 1: If $\left( \dfrac{M_{y}-N_{x}}{N} \right)$ is a function of $x$ only, say $\xi(x)$. Then, the integrating factor is given by:

\[\mu(x)=e^{\int \xi(x) d x}\]

Case 2: If $\left( \dfrac{M_{y}-N_{x}}{-M} \right)$ is a function of $y$ only, say $\psi(y)$. Then, the integrating factor is given by:

\[\mu(y)=e^{\int \psi(y) d y}\]

Orthogonal Trajectory

The orthogonal trajectory for a given curve $y=f(x,c)$ can be found by forming the differential equation (eliminating $c$) and replacing $dy/dx$ by $-\dfrac{1}{dy/dx}$

Equations Of First Order But Not Of First Degree

Such differential equations contain only the first order differential coefficient $\dfrac{dy}{dx}$ but will occur in a degree higher than one.

These equations can be solved by factorization and solving for $\dfrac{dy}{dx}$.

Clairaut’s Equation and Singular Solution

An equation of the form $y=px+f(p)$, where $p=\dfrac{dy}{dx}$ is called Clairaut’s Equation.

Differentiating this equation, we get $\mathrm{p}^{\prime}\left[\mathrm{x}+\mathrm{f}^{\prime}(\mathrm{p})\right]=0$, which implies that either $p'=0$, or $x+f'(p)=0$.

Therefore, the first equation $p'=0$ gives the general solution of the Clairaut’s Equation as $y=cx+f(c)$ and the second equation $x+f'(p)=0$, combined with the Clairaut’s Equation $y=px+f(p)$, gives the singular solution (by eliminating $p$) which is free from any arbitrary constant.

PYQs

Formulation Of Differential Equations

1) Find the differential equation representing the entire circle in the $xy-plane$.

[2017, 10M]

2) Consider the differential equation $y^{\prime}=\alpha y, x>0$ where $\alpha$ is a constant. Show that:
i) If $\phi(x)$ is any solution and $\psi(x)=\phi(x) e^{-\alpha x}$, then $\psi(x)$ is a constant. ii) If $\alpha<0$, then every solution tends to zero as $x \rightarrow \infty$.

[2010, 12M]

3) Find the differential equation of the family of circles in the $xy-plane$ passing through $(-1,1)$ and $(1,1)$.

[2009, 20M]

4) A continuous function $y(t)$ satisfies the differential equation:

\[\dfrac{d y}{d x}=\left\{\begin{array}{ll}{1+e^{1 - t},} & {0 \leq t< 1} \\ {2+2 t-3 t^{2},} & {1 \leq t< 5}\end{array}\right.\]

If $y(0)=-e$, find $y(2)$.

[2001, 12M]

Equations Of First Order And First Degree

1) Determine the complete solution of the differential equation

\[\dfrac{d^2y}{dx^2}-4\dfrac{dy}{dx}+4y=3x^2e^{2x} \sin x\]

[2019, 10M]

2) Solve the differential equation

\[(2y \sin x+3y^4 \sin x \cos x)dx-(4y^3 \cos^2x+ \cos x)dy=0\]

[2019, 10M]

3) Solve the following simultaneous liner differential equations:
$(D+1) y=z+e^{x}$ and $(D+1) z=y+e^{x}$
where $y$ and $z$ are functions of independent variable $x$ and $D \equiv \dfrac{d}{d x}$.

[2017, 8M]

4) If the growth rate of the population of bacteria at time $t$ is proportional to the amount present at the time $t$ and population doubles in one week, then how much bacteria can be expected after 4 weeks?

[2017, 8M]

5) Solve:

\[\dfrac{dy}{dx} = \dfrac{1}{1+x^2}(e^{\tan^{-1}x}-y)\]

[2016, 10M]

6) Solve the differential equation:

$\left(2 x y^{4} e^{y}+2 x y^{3}+y\right) d x+\left(x^{2} y^{4} e^{y}-x^{2} y^{2}-3 x\right) d y=0$.

[2015, 10M]

7) Solve by the method of variation of parameters: $\dfrac{d y}{d x}-5 y=\sin x$.

[2014, 10M]

8) If $y$ is a function of $x$ such that the differential coefficient $\dfrac{d y}{d x}$ is equal to $\cos (x+y)+\sin (x+y)$, find out a relation between $x$ and $y$$ which is free from any derivative/ differential.

[2013, 10M]

9) Solve the differential equation $\left(5 x^{3}+12 x^{2}+6 y^{2}\right) d x+6 x y d y=0$

[2013, 15M]

10) Solve $\dfrac{d y}{d x}=\dfrac{2 x y e^{(x / y)^{2}}}{y^{2}\left(1+e^{(x / y)^{2}}\right)+2 x^{2} e^{(x / y)^{2}}}$.

[2012, 12M]

11) Obtain the solution of the ordinary differential equation $\dfrac{d y}{d x}=(4 x+y+1)^{2}$, if $y(0)=1$

[2011, 10M]

12) Show that the set of solutions of the homogeneous linear differential equation

\[y'+p(x)y=0\]

on an interval $I=[a,b]$ forms a vector subspace $W$ of the real vector space of continuous functions on $I$. What is the dimension of $W$?

[2010, 20M]

13) Solve: $\dfrac{d y}{d x}=\dfrac{y^{2}(x-y)}{3 x y^{2}-x^{2} y-4 y^{3}}, y(0)=1$

[2009, 20M]

14) Solve the differential equation $y d x+\left(x+x^{3} y^{2}\right) d y=0$

[2008, 12M]

15) Find the solution of the equation $\dfrac{d y}{y}+x y^{2} d x=-4 x d x$

[2007, 12M]

16) Solve the differential equation: $\left(x y^{2}+e^{-\dfrac{1}{x^{3}}}\right) d x-x^{2} y d y=0$.

[2006, 12M]

17) Solve: $\left(1+y^{2}\right)+\left(x-e^{-\tan ^{-1} y}\right) \dfrac{d y}{d x}=0$.

[2006, 15M]

18) Solve: $x y \dfrac{d y}{d x}=\sqrt{\left(x^{2}-y^{2}-x^{2} y^{2}-1\right)}$.

[2005, 12M]

19) Solve: $y\left(x y+2 x^{2} y^{2}\right) d x+x\left(x y-x^{2} y^{2}\right) d y=0$.

[2004, 12M]

20) Solve: $x \dfrac{d y}{d x}+y \log y=x y e^{x}$.

[2003, 12M]

21) Solve: $x \dfrac{d y}{d x}+3 y=x^{3} y^{2}$.

[2002, 12M]

22) Solve: $\dfrac{d y}{d x}=\dfrac{x+y+4}{x-y-6}$.

[2002, 15M]

23) Solve: $\dfrac{d y}{d x}+\dfrac{y}{x} \log _{e} y=\dfrac{y\left(\log _{e} y\right)^{2}}{x^{2}}$.

[2001, 15M]

Integrating Factor

1) Find $f(y)$ such that $(2xe^y+3y^2)dy+(3x^2+f(y))dx=0$ is exact and hence solve.

[2018, 13M]

2) Find $\alpha$ and $\beta$ such that $x^{\alpha}y^{\beta}$ is an integrating factor of $(4y^2+3xy)dx-(3xy+2x^2)dy=0$ and solve the equation.

[2018, 12M]

3) Solve $\left\{y(1-x \tan x)+x^{2} \cos x\right\} d x-x d y=0$.

[2016, 10M]

4) Solve the differential equation: $x \cos x \dfrac{d y}{d x}+y(x \sin x+\cos x)=1$.

[2015, 10M]

5) Find the constant $a$ so that $(x+y)^{a}$ is the integrating factor of $\left(4 x^{2}+2 x y+6 y\right) d x+\left(2 x^{2}+9 y+3 x\right) d y=0$ and hence solve the differential equation.

[2015, 12M]

6) Justify that a differential equation of the form: $\left[y+x f\left(x^{2}+y^{2}\right)\right] d x+\left[y f\left(x^{2}+y^{2}\right)-x\right] d y=0$ where $f\left(x^{2}+y^{2}\right)$ is an arbitrary function of $\left(x^{2}+y^{2}\right)$, is not an exact differential equation and $\dfrac{1}{x^{2}+y^{2}}$ is an integrating factor for it. Hence solve this differential equation for $f\left(x^{2}+y^{2}\right)=\left(x^{2}+y^{2}\right)^{2}$.

[2014, 10M]

7) Find the sufficient condition for the differential equation $M(x, y) d x+N(x, y) d y=0$ to have an integrating factor as a function of $(x+y)$. What will be the integrating factor in that case? Hence find the integrating factor for the differential equation of $\left(x^{2}+x y\right) d x+\left(y^{2}+x y\right) d y=0$ and solve it.

[2014, 15M]

8) Show that the differential equation $(2 x y \log y) d x+\left(x^{2}+y^{2} \sqrt{y^{2}+1}\right) d y=0$ is not exact. Find an integrating factor and hence, the solution of the equation.

[2012, 20M]

9) Show that the differential equation $\left(3 y^{2}-x\right)+2 y\left(y^{2}-3\right) y^{\prime}=0$ admits an integrating factor which is a function of $\left(x+y^{2}\right)$. Hence solve the equation.

[2010, 12M]

10) Verify that $\dfrac{1}{2}(M x+N y) d\left[\log _{e}(x y)\right]+\dfrac{1}{2}(M x-N y) d\left[\log _{e}(x / y)\right]=M d x+N d y$. Hence show that:

i) If the differential equation $M d x+N d y=0$ is homogeneous, then $(M x+N y)$ is an integrating factor unless $M x+N y \equiv 0$.

ii) If the differential equation $M d x+N d y=0$ is not exact but is of the form $f_{1}(x y) y d x+f_{2}(x y) x d y=0$ then $(M x-N y)^{-1}$ is an integrating factor unless $M x+N y \equiv 0$.
[2010, 20M]

11) Solve the ordinary differential equation $\cos 3 x \dfrac{d y}{d x}-3 y \sin 3 x=\dfrac{1}{2} \sin 6 x+\sin ^{2} 3 x$, $0 < x < \dfrac{\pi}{2}$.

[2007, 15M]

12) Find the solution of the following differential equation $\dfrac{d y}{d x}+y \cos x=\dfrac{1}{2} \sin 2 x$.

[2004, 12M]

13) Find the value of constant $\lambda$ such that the following differential equation becomes exact.
$\left(2 x e^{y}+3 y^{2}\right) \dfrac{d y}{d x}+\left(3 x^{2}+\lambda e^{y}\right)=0$. Further, for this value of $\lambda$, solve the equation.

[2002, 15M]

Orthogonal Trajectory

1) Show that the family of parabolas $y^{2}=4 c x+4 c^{2}$ is self orthogonal.

[2016, 10M]

2) Find the curve for which the part of the tangent cut-off by the axes is bisected at the point of tangency.

[2014, 10M]

3) Obtain the equation of the orthogonal trajectory of the family of curves represented by $r^{n}=a \sin n \theta,(r, \theta)$ being the plane polar coordinates.

[2013, 10M]

4) Find the orthogonal trajectories of the family of curves $x^{2}+y^{2}=a x$.

[2012, 12M]

5) Determine the orthogonal trajectory of a family of curves represented by the polar equation $r=a(1-\cos \theta),(r, \theta)$ being the plane polar coordinates of any point.

[2011, 10M]

6) Find the family of curves whose tangents form an angle $\dfrac{\pi}{4}$ with the hyperbolas $x y=c, c>0$.

[2006, 12M]

7) Find the orthogonal trajectory of the family of co-axial circles $x^{2}+y^{2}+2 g x+c=0$, where $g$ is the parameter.

[2005, 12M]

8) Show that the orthogonal trajectory of a system of confocals ellipses is self orthogonal.

[2003, 12M]

Equations Of First Order But Not Of First Degree

1) Solve the differential equation $x=p y-p^{2}$ where $p=\dfrac{d y}{d x}$.

[2015, 13M]

2) Solve the equation $y-2 x p+y p^{2}=0$, where $p=\dfrac{d y}{d x}$.

[2008, 15M]

Clairaut’s Equation and Singular Solution

1) Solve:

\[\left( \dfrac{dy}{dx} \right)^2y+2\dfrac{dy}{dx}-y=0\]

[2018, 13M]

2) Consider the differential equation $x y p^{2}-\left(x^{2}+y^{2}-1\right) p+x y=0$ where $p=\dfrac{d y}{d x}$ substituting $u=x^{2}$ and $v=y^{2}$ reduce the equation to Clairaut’s form in terms of $u$, $v$ and $p^{\prime}=\dfrac{d v}{d u}$ hence or otherwise solve the equation.

[2017, 10M]

3) Obtain Clairaut’s form of the differential equation $\left(x \dfrac{d y}{d x}-y\right)\left(y \dfrac{d y}{d x}+x\right)=a^{2} \dfrac{d y}{d x}$ Also find its general solution.

[2011, 15M]

4) Determine the general and singular solutions of the equation

\[y=x\dfrac{dy}{dx}+a\dfrac{dy}{dx} \left[ 1+ \left( \dfrac{dy}{dx} \right)^2 \right]^{-\frac{1}{2}},\]

$a$ being a constant.

[2007, 15M]

5) Solve the equation

\[x^2p^2+yp(2x+y)+y^2=0\]

using the substitution $y=u$ and $xy=v$ and find its singular solution, where $p=\dfrac{dy}{dx}$

[2006, 15M]

6) Solve the differential equation: $\left(x^{2}+y^{2}\right)(1+p)^{2}-2(x+y)(1+p)(x+y p)+(x+y p)^{2}=0$ where $p=\dfrac{d y}{d x}$, by reducing it to Clairaut’s form by using suitable substitution.

[2005, 15M]

7) Reduce the equation $(p x-y)(p y+x)=2 p$, where $p=\dfrac{d y}{d x}$ to Clairaut’s equation and hence solve it.

[2004, 15M]

8) Solve the differential equation $\left(p x^{2}+y^{2}\right)(p x+y)=(P+1)^{2}$, where $p=\dfrac{a y}{d x}$, by reducing it to Clairaut’s form using suitable substitutions.

[2003, 15M]

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