2022-03-25

Differential equations
${\left(x+1\right)}^{2}y{}^{″}+\left(x+1\right){y}^{\prime }+y={x}^{2}+2\mathrm{sin}\left(\mathrm{ln}\left(x+1\right)\right),y\left(0\right)=\frac{1}{5},{y}^{\prime }\left(0\right)=2$

Demetrius Kaufman

Step 1
$y{}^{″}+\frac{{y}^{\prime }}{\left(x+1\right)}+\frac{y}{{\left(x+1\right)}^{2}}=\frac{{x}^{2}+2\mathrm{sin}\left(\mathrm{ln}\left(x+1\right)\right)}{{\left(x+1\right)}^{2}}$
First of all , I found the equation solution of $y{}^{″}+\frac{{y}^{\prime }}{\left(x+1\right)}+\frac{y}{{\left(x+1\right)}^{2}}=0$
$y={c}_{1}\mathrm{cos}\left(\mathrm{ln}\left(x+1\right)\right)+{c}_{2}\mathrm{sin}\left(\mathrm{ln}\left(x+1\right)\right)$
I try to solve this ode using the variation of parameters theorem
Get this system equation:
1) ${c}_{1}^{\prime }\mathrm{cos}\left(\mathrm{ln}\left(x+1\right)\right)+{c}_{2}^{\prime }\mathrm{sin}\left(\mathrm{ln}\left(x+1\right)\right)=0$
2) $-{c}_{1}^{\prime }\mathrm{sin}\left(\mathrm{ln}\left(x+1\right)\right)+{c}_{2}^{\prime }\mathrm{cos}\left(\mathrm{ln}\left(x+1\right)\right)={x}^{2}+2\mathrm{sin}\left(\mathrm{ln}\left(x+1\right)\right)$
Multiply 1) by $\mathrm{sin}\left(\mathrm{ln}\left(x+1\right)\right)$, 2) by $\frac{\mathrm{cos}\left(\mathrm{ln}\left(x+1\right)\right)}{x+1}$.
Step 2
${c}_{2}^{\prime }=\frac{\mathrm{cos}\left(\mathrm{ln}\left(x+1\right)\right)\left[{x}^{2}+2\mathrm{sin}\left(\mathrm{ln}\left(x+1\right)\right)\right]}{\left(x+1\right)}$
I do not know how I get ${c}_{2}$ by an integral ?

undodaonePvopxl24

Step 1
The two equations you have using variation of parameters is incorrect. Given a second-order linear inhomogeneous DE  with homogeneous solution ${y}_{c}\left(x\right)={c}_{1}{y}_{1}\left(x\right)+{c}_{2}{y}_{2}\left(x\right)$, a particular solution ${y}_{p}$ is given by ${y}_{p}\left(x\right)={u}_{1}\left(x\right){y}_{1}\left(x\right)+{u}_{2}\left(x\right){y}_{2}\left(x\right)$, where ${u}_{1}$ and ${u}_{2}$ satisfy

$\begin{array}{rl}{u}_{1}^{\prime }{y}_{1}+{u}_{2}^{\prime }{y}_{2}& =0\\ {u}_{1}^{\prime }{y}_{1}^{\prime }+{u}_{2}^{\prime }{y}_{2}^{\prime }& =f\left(x\right).\end{array}$

Let .

We have that
Step 2
Multiplying (1) by $\mathrm{sin}\left(\mathrm{ln}\left(x+1\right)\right)$, (2) by $\left(x+1\right)\mathrm{cos}\left(\mathrm{ln}\left(x+1\right)\right)$, and adding the two resulting equations, we get

Step 3
Using the substitution $z=\mathrm{ln}\left(x+1\right)$, we see that

Computing this integral is a standard exercise now. Can you take it from here?

Do you have a similar question?