General solution of the given differential equation is given by:
[tex]$$u = {e^{mx}}(c_1{e^{k_1}x} + c_2{e^{k_2}x})y(x) + {e^{mx}}(c_1 \cos (\frac{{\sqrt {2\mu - {\mu ^2} - 36{m^2}} }}{4}x) + c_2 \sin (\frac{{\sqrt {2\mu - {\mu ^2} - 36{m^2}} }}{4}x))y(x)$$[/tex]
Where c1 and c2 are arbitrary constants.
i) To find the general solutions of the given differential equation, we proceed as follows:
[tex]$$uxx - 4u_{x} + u_{y} + 2u = 9 \sin (3x - y) + 19 \cos (3x - y)$$[/tex]
Using the characteristic equation: [tex]$$r^2 - 4r + 1 = 0$$[/tex]
Solving it, we get
$$r = \frac{{4 \pm \sqrt {14} }}{2} = 2 \pm \sqrt 3 $$
Therefore, the complementary function is given by:
[tex]$$u_{c} = {e^{2x}}(c_1 \cos (\sqrt 3 x) + c_2 \sin (\sqrt 3 x))$$[/tex]
Particular integral: To find the particular integral, we follow the steps as mentioned below: hom*ogeneous equation:
[tex]$$u_{xx} - 4u_{x} + u_{y} + 2u = 0$$[/tex]
Now, consider a particular integral of the form:
[tex]$$u_{p} = (A\sin (3x - y) + B\cos (3x - y))$$[/tex]
Differentiating once with respect to x:
[tex]$$u_{px} = 3A\cos (3x - y) - 3B\sin (3x - y)$$[/tex]
Differentiating twice with respect to x:
[tex]$$u_{pxx} = - 9A\sin (3x - y) - 9B\cos (3x - y)$$[/tex]
Differentiating with respect to y:
[tex]$$u_{py} = - A\cos (3x - y) - B\sin (3x - y)$$[/tex]
Substituting the above values in the given equation, we get:
[tex]$$ - 9A\sin (3x - y) - 9B\cos (3x - y) - 4(3A\cos (3x - y) - 3B\sin (3x - y)) + ( - A\cos (3x - y) - B\sin (3x - y)) + 2(A\sin (3x - y) + B\cos (3x - y)) = 9\sin (3x - y) + 19\cos (3x - y) $$[/tex]
Simplifying the above equation, we get:
[tex]$$[ - 6A - B + 2A + 2B]\cos (3x - y) + [ - 6B + A + 2A + 2B]\sin (3x - y) = 9\sin (3x - y) + 19\cos (3x - y) + 9A\sin (3x - y) + 9B\cos (3x - y) $$[/tex]
Comparing coefficients of [tex]$\sin (3x - y)$ and $\cos (3x - y)$, we get:$$ - 7A + 4B = 0\hspace{0.5cm}(1)$$$$4A + 23B = 19\hspace{0.5cm}(2)$$[/tex]
Solving equations (1) and (2), we get:
[tex]$$A = \frac{{23}}{{103}}$$\\[/tex]
Substituting the value of A in equation (1), we get:
[tex]$$B = \frac{{161}}{{309}}$$[/tex]
Therefore, the particular integral is given by:
[tex]$$u_{p} = \frac{{23}}{{103}}\sin (3x - y) + \frac{{161}}{{309}}\cos (3x - y)$$[/tex]
The general solution of the given differential equation is given by:
[tex]$$u = u_{c} + u_{p}$$$$u = {e^{2x}}(c_1 \cos (\sqrt 3 x) + c_2 \sin (\sqrt 3 x)) + \frac{{23}}{{103}}\sin (3x - y) + \frac{{161}}{{309}}\cos (3x - y)$$ii) $$4u_{xx} + 4u_{x} + u + 12\mu x + 6\mu y + 9u = 0$$[/tex]
Let [tex]$$u = {e^{mx}}y(x)$$[/tex]
Differentiating w.r.t x, we get:
[tex]$$u_{x} = m{e^{mx}}y + {e^{mx}}y'$$[/tex]
Differentiating again w.r.t x, we get:
[tex]$$u_{xx} = m^2{e^{mx}}y + 2m{e^{mx}}y' + {e^{mx}}y''$$[/tex]
Substituting the above values, we get:
[tex]$$4{e^{mx}}[m^2y + 2my' + y''] + 4{e^{mx}}[my + y'] + {e^{mx}}y + 12\mu x + 6\mu y + 9{e^{mx}}y = 0$$[/tex]
Simplifying the above equation, we get:
[tex]$$4{e^{mx}}y'' + (8m + 4\mu ){e^{mx}}y' + (4m^2 + 9){e^{mx}}y + 12\mu x = 0$$$$4y'' + (8m + 4\mu )y' + (4m^2 + 9)y + 12\mu xy = 0$$[/tex]
Characteristic equation:
[tex]$$4r^2 + (8m + 4\mu )r + (4m^2 + 9) = 0$$[/tex]
Solving the above equation, we get:
[tex]$$r = \frac{{ - 2m - \mu \pm \sqrt {{{(2m + \mu )}^2} - 4(4{m^2} + 9)} }}{8}$$Case (i):$$r = \frac{{ - 2m - \mu + \sqrt {{{(2m + \mu )}^2} - 4(4{m^2} + 9)} }}{8} = {k_1}$$$$r = \frac{{ - 2m - \mu - \sqrt {{{(2m + \mu )}^2} - 4(4{m^2} + 9)} }}{8} = {k_2}$$[/tex]
The complementary function is given by:
[tex]$$u_{c} = {e^{mx}}(c_1{e^{k_1}x} + c_2{e^{k_2}x})y(x)$$Case (ii):$$r = \frac{{ - 2m - \mu + \sqrt {{{(2m + \mu )}^2} - 4(4{m^2} + 9)} }}{8}$$$$r = \frac{{ - 2m - \mu - \sqrt {{{(2m + \mu )}^2} - 4(4{m^2} + 9)} }}{8}$$[/tex]
Therefore, the complementary function is given by:
[tex]$$u_{c} = {e^{mx}}(c_1 \cos (\frac{{\sqrt {2\mu - {\mu ^2} - 36{m^2}} }}{4}x) + c_2 \sin (\frac{{\sqrt {2\mu - {\mu ^2} - 36{m^2}} }}{4}x))y(x)$$[/tex]
General solution:
The general solution of the given differential equation is given by:
[tex]$$u = {e^{mx}}(c_1{e^{k_1}x} + c_2{e^{k_2}x})y(x) + {e^{mx}}(c_1 \cos (\frac{{\sqrt {2\mu - {\mu ^2} - 36{m^2}} }}{4}x) + c_2 \sin (\frac{{\sqrt {2\mu - {\mu ^2} - 36{m^2}} }}{4}x))y(x)$$[/tex]
Where c1 and c2 are arbitrary constants.
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