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If someone could complete all of these it would be greatly appreciated.

1) Determine if the undetermined coefficients method can be used for the following ODEs. For the one(s) that can be solved with undetermined coefficients, determine a suitable form for \( y_{p} \) if the method of undetermined coefficients is to be used for the following ODEs with following steps: i) find the basis of the homogeneous part of the ODE, ii) look up Table \( 2.1 \) on page 80 and iii) apply modification and sum rule if necessary. No need to evaluate constants. (14 points) a) \( y^{\prime \prime}+4 y=x(1+\sin 2 x) \). b) \( y^{\prime \prime}+16 y=x^{2} \sin 4 x+(6 x+7) \cos 4 x \) c) \( 3 y^{\prime \prime}+149 y^{\prime}-50 y=\frac{1}{x^{2}} \) d) \( y^{\prime \prime}+6 y^{\prime}+9 y=100 \sinh 3 x \) (See Lecture 8, pp. 8-12, 14) 2) Use the variation of parameters to solve \( y^{\prime \prime}-4 y^{\prime}+4 y=12 e^{2 x} / x^{3} \), i.e. find the general solution. (10 points) (See Lecture 9 pp. 6-8 and variation of parameters examples 1 and 2 at Lecture 09 and examples on Canvas) 3) Use the variation of parameters to solve \( y^{\prime \prime}+4 y=4 \sec ^{2} 2 x \), i.e. find the general solution. (10 points) (See Lecture 9 pp. 6-8 and variation of parameters examples 1 and 2 at Lecture 09 and examples on Canvas) 4) Find the roots of the denominator of each \( F(s) \) and identify its cases for partial fraction expansion, e.g. cases I, II, III, and IV and the forms of partial fractions. Notice that \( F(s) \) may belong to more than one case (see partial fraction examples 1 to 4 at Lecture 09 and examples, pp. 28-30 of lecture 9) a) \( F(s)=\frac{s}{s^{2}-1.5 s+0.5} \cdot \) (3 points) b) \( F(s)=\frac{s^{2}}{(s+1)\left(s^{2}-2 s+1\right)} \cdot(4 \) points \( ) \) c) \( F(s)=\frac{s+1}{\left(s^{2}+1\right)\left(s^{2}+9\right)^{2}} \) (4 points) d) \( F(s)=\frac{s^{2}}{\left(s^{2}-1\right)^{2}(s-1)} \) (4 points) e) \( F(s)=\frac{4 s^{4}-18}{2 s\left(2 s^{2}+4 s+10\right)\left(s^{2}+4 s+13\right)} \) (4 points \( ) \)

1) a) y?+4y=x+xsin?(2x) (i) Auxiliary equation m2+4=0 m=±2i Hence basis for homogeneous part is {cos?(2x),sin?(2x)} (ii) If we look at the non homogen