We will cover following topics

ODEs With Variable Coefficients
- Euler-Cauchy Equation
- Determination Of Complete Solution When One Solution Is Known
PYQs

ODEs With Variable Coefficients

Equations of the form $\frac{d^{2} y}{d x^{2}} + p (x) \frac{d y}{d x} + q (x) y = r (x)$ are known as second order linear equations with variable coefficients.

Euler-Cauchy Equation

An equation of the form $a_{n} x^{n} y^{(n)} (x)$ + $a_{n - 1} x^{n - 1} y^{(n - 1)} (x)$ + $\dots$ + $a_{0} y (x) = 0$ is known as Euler-Cauchy equation.

The equation is converted into an equation with constant coefficients by first substituting $x = e^{u}$ .

The equation can then be solved by first finding the complementary function using the homogeneous equation and then finding the particular integral.

The complete solution is written as:

y = c o m p l e m e n t a r y f u n c t i o n + p a r t i c u l a r i n t e g r a l

Determination Of Complete Solution When One Solution Is Known

If a particular solution $y_{1} (x)$ of the homogeneous linear second order equation is known, the original equation can be converted to a linear first order equation using the substitution $y = y_{1} (x) z (x)$ and the subsequent replacement $z^{'} (x) = u$ .

Another way to reduce the order is based on the Liouville formula. In this case, a particular solution $y_{1} (x)$ must also be known.

PYQs

ODEs with Variable Coefficients

1) Solve the differential equation

\frac{d^{2} y}{d x^{2}} + (3 s i n x - c o t x) \frac{d y}{d x} + 2 y s i n^{2} x = e^{- c o s x} s i n^{2} x

[2019, 10M]

2) Solve the differential equation: $x \frac{d^{2} y}{d x^{2}} - \frac{d y}{d x} - 4 x^{3} y = 8 x^{3} \sin (x^{2})$

[2017, 8M]

3) Solve the following differential equation: $x \frac{d^{2} y}{d x^{2}} - 2 (x + 1) \frac{d y}{d x} + (x + 2) y = (x - 2) e^{2 x}$ , when $e^{x}$ is a solution to its corresponding homogeneous differential equation.

[2014, 15M]

4) Solve the ordinary differential equation $x (x - 1) y^{''} - (2 x - 1) y^{'} + 2 y = x^{2} (2 x - 3)$ .

[2012, 20M]

5) Solve: $(x + 2) \frac{d^{2} y}{d x^{2}} - (2 x + 5) \frac{d y}{d x} + 2 y = (x + 1) e^{x}$ .

[2004, 15M]

6) Solve the folllowing differential equation:

(1 - x^{2}) \frac{d^{2} y}{d x^{2}} - 4 x \frac{d y}{d x} - (1 + x^{2}) y = x

[2004, 15M]

7) Solve ${(1 + x)}^{2} y^{''} + (1 + x) y^{'} + y = \sin 2 [\log (1 + x)]$ .

[2003, 15M]

Euler-Cauchy Equation

1) Obtain the singular solution of the differential equation

(\frac{d y}{d x})^{2} (\frac{y}{x})^{2} c o t^{2} α - 2 (\frac{d y}{d x}) (\frac{y}{x}) + (\frac{d y}{d x})^{2} c o s e c^{2} α = 1

Also find the complete primitive of the given differential equation. Give the geometrical interpretations of the complete primitive and singular solution.

[2019, 15M]

2) Find the linearly independent solutions of the corresponding homogeneous differential equation of the equation $x^{2} y^{″} - 2 x y^{'} + 2 y = x^{3} \sin x$ and then find the general solution of the given equation by the method of variation of parameters.

[2019, 15M]

3) Solve:

(1 + x)^{2} y^{″} + (1 + x) y^{'} + y = 4 c o s (l o g (1 + x))

[2018, 13M]

4) Solve $x^{4} \frac{d^{4} y}{d x^{4}} + 6 x^{3} \frac{d^{3} y}{d x^{3}} + 4 x^{2} \frac{d^{2} y}{d x^{2}} - 2 x \frac{d y}{d x} - 4 y = x^{2} + 2 \cos (\log_{e} x)$ .

[2015, 13M]

5) Solve the differential equation: $x^{3} \frac{d^{3} y}{d x^{3}} + 3 x^{2} \frac{d^{2} y}{d x^{2}} + x \frac{d y}{d x} + 8 y = 65 \cos (\log_{e} x)$ .

[2014, 20M]

6) Solve the differential equation $x^{3} y^{''} - 3 x^{2} y^{'} + x y = \sin (\ln x) + 1$ .

[2008,15M]

7) Solve the equation $2 x^{2} \frac{d^{2} y}{d x^{2}} + 3 x \frac{d y}{d x} - 3 y = x^{3}$ .

[2007, 15M]

8) Solve the differential equation $x^{2} \frac{d^{3} y}{d x^{3}} + 2 x \frac{d^{2} y}{d x^{2}} + 2 \frac{y}{x} = 10 (1 + \frac{1}{x^{2}})$ .

[2006, 15M]

9) Solve the differential equation: $[(x + 1)^{4} D^{3} + 2 (x + 1)^{3} D^{2} - (x + 1)^{2} D + (x + 1)] y = \frac{1}{(x + 1)}$ .

[2005, 15M]

10) Find the values of $λ$ for which all solutions of $x^{2} \frac{d^{2} y}{d x^{2}} + 3 x \frac{d y}{d x} - λ y = 0$ tend to zero as $x \to \infty$ .

[2002, 12M]

11) Solve: $x^{2} \frac{d^{2} y}{d x^{2}} - x \frac{d y}{d x} - 3 y = x^{2} \log_{e} x$ .

[2001, 12M]

Determination of Complete Solution when One Solution is known

1) Using the method of variation of parameters, solve the differential equation $(D^{2} + 2 D + 1) y = e^{- x} \log (x), [D \equiv \frac{d}{d x}]$ .

[2016, 15M]

2) Using the method of variation of parameters, solve the differential equation $\frac{d^{2} y}{d x^{2}} + a^{2} y = \sec a x$ .

[2013, 15M]

3) Using the method of variation of parameters, solve the second order differential equation $\frac{d^{2} y}{d x^{2}} + 4 y = \tan 2 x$ .

[2011, 15M]

4) Use the method of variation of parameters to find the general solution of $x^{2} y^{''} - 4 x y^{'} + 6 y = - x^{4} \sin x$ .

[2008, 12M]

5) Use the method of variation of parameters to find the general solution of the equation $\frac{d^{2} y}{d x^{2}} + 3 \frac{d y}{d x} + 2 y = 2 e^{x}$ .

[2007, 15M]

6) Solve the differential equation $(D^{2} - 2 D + 2) y = e^{x} \tan x, D \equiv \frac{d y}{d x}$ by the method of variation of parameters.

[2006, 15M]

7) Solve the differential equation $(\sin x - x \cos x) y^{''} - x \sin x y^{'} + y \sin x = 0$ given that $y = \sin x$ is a solution of this equation.

[2005, 15M]

8) Solve the differential equation $x^{2} y^{''} - 2 x y^{'} + 2 y = x \log x, x > 0$ by variation of parameters.

[2005, 15M]

9) Solve the differential equation

x^{2} y^{''} - 4 x y' + 6 y = x^{4} \sec^{2} x

by variation of parameters.

[2003, 15M]

10) Using the method of variation of parameteres, find the solution of

\frac{d^{2} y}{d x^{2}} - 2 \frac{d y}{d x} + y = x e^{x} \sin x

with $y (0) = 0$ and ${(\frac{d y}{d x})}_{x = 0} = 0$

[2002, 15M]

11) Using the method of variation of parameters, solve:

\frac{d^{2} y}{d x^{2}} + 4 y = 4 \tan 2 x

[2001, 15M]

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