 # Linearizing y′′+\sin(2x+\cos(2y′+y))+1- \sin(y+3y′)=0 Mary Buchanan 2022-01-20 Answered
Linearizing $y\prime \prime +\mathrm{sin}\left(2x+\mathrm{cos}\left(2y\prime +y\right)\right)+1-\mathrm{sin}\left(y+3y\prime \right)=0$
You can still ask an expert for help

## Want to know more about Laplace transform?

• Live experts 24/7
• Questions are typically answered in as fast as 30 minutes
• Personalized clear answers

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it Tiefdruckot
The easiest linearization I can think of in this case is to use is small enough. In your case, these assumptions boil down to making the assumption that y, ${y}^{\prime }\approx 0$.
With these assumptions, your differential equation now becomes $y{}^{″}+\mathrm{sin}\left(2x+1\right)+1-\left(y+3{y}^{\prime }\right)=0$
which in turn becomes
$y{}^{″}-3{y}^{\prime }-y=-\left(1+\mathrm{sin}\left(2x+1\right)\right)$
Now you can use your differential equation tricks to solve the above equation to get
$y\left(x\right)={c}_{1}\mathrm{exp}\left(\left(\frac{3-\sqrt{13}}{2}\right)x\right)+{c}_{2}\mathrm{exp}$
$\left(\left(\frac{3+\sqrt{13}}{2}\right)x\right)+\frac{5\mathrm{sin}\left(2x+1\right)-6\mathrm{cos}\left(2x+1\right)+61}{61}$