Subject : Applied Mathematics 2

Topic : Linear Differential Equations With Constant Coefficients and Variable Coefficients Of Higher Order

Difficulty: High

Solution:

$(1+x)^{2} \frac{d^{2} y}{d x^{2}}+(1+x) \frac{d y}{d x}+y=4 \cos (\log (1+x))$

Put $\quad x+1=v \quad \Rightarrow \quad \frac{d v}{d x}=1$

$\frac{d y}{d x}=\frac{d y}{d v}$

The given ean changes to.

$v^{2} \frac{d^{2} y}{d v^{2}}+v \frac{d y}{d v}+y=4 \cos \log v$

Now put $\log v=z \quad \therefore v=e^{z}$

$[D(D-1)+D+1] y=4 \cos z$

$\therefore\left(D^{2}+1\right) y=4 \cos z$

For complementary solution,

$f(D)=0$

$\therefore\left(D^{2}+1\right)=0$

Roots are : $\mathrm{i},-\mathrm{i}$

The complementary solution of given diff. eqn is,

$\therefore y_{c}=c_{1} \cos z+c_{2} \sin z$

For particular integral,

$y_{p}=\frac{1}{f(D)} X=\frac{1}{\left(D^{2}+1\right)} 4 \cos z=4 \frac{z}{2} \sin z=2 \mathbf{z} \sin \mathbf{z}$

$\therefore y_{p}=2 z \sin z$

The general solution of given diff. eqn is given by,

$y_{g}=y_{c}+y_{p}=c_{1} \cos z+c_{2} \sin z+2 z \sin z$

Resubstitute z and v

$y_{g}=c_{1} \cos [\log (x+1)]+c_{2} \sin [\log (1+x)]+2 \log (1+x) \sin [\log (1+x)]$