Question

\[3y'+2y=x^3sin(\pi*x)\]

Answer 1

Give a min plzz, i Think we need to use the integrating factor here

Answer 2

We can write the equation as - \[y ^{'} + (2/3)y = (1/3)x ^{2}\sin \pi x\]
Hence the integrating factor becomes \[\exp[\int\limits_{}^{}(2/3)dx] = e ^{(2/3)x}\]

Multiplying both sides by this factor of the equation we get -

\[e ^{(2/3)x}y {^'} + (2/3)ye ^{(2/3)x} = (1/3)e ^{(2/3)x}x ^{2}\sin \pi x\]

Now the left hand side is the derivative of

Answer 3

Continuing from earlier post -
Left hand side is \[d/dx(ye ^{(2/3)x})\]

Hence the solution becomes -
\[ye ^{(2/3)x} = (1/3)\int\limits_{}^{}e ^{(2/3)x} x ^{2} \sin \pi x dx\]

Right hand side we need to use Integration by parts, using x^2 as first function, sin pi x as the seond function and the exponential as the third function

Answer 4

Can you do the integration by parts?

Answer 5

When I use the method of udetermined coefficients I get;
\[\huge y_g = c_1e^{-\frac{2}{3}x}+\left(\frac{1}{2}x^3-\frac{9}{4}x^2+\frac{27}{4}x-\frac{81}{8} \right) \]\[\huge*\left(\frac{2}{9\pi^2+4}\text{sin}(\pi*x)-\frac{3\pi}{9\pi^2+4}\text{cos}(\pi*x) \right)\]