Q1- In the following problems state the order of the given ordinary differential equations and determine whether the equations are linear or nonlinear ( 6 marks). 1. (1x)y4xy+5y=cosx 2. xdx3d3y(dxdy)4+y=0 3. t5y(4)t3y+6y=0 4. dr2d2u+drdu+u=cos(r+u) 5. dx2d2y=1+(dxdy)2 6. dt2d2R=R2k Q2- Using the "integrating factor" method find the general solution to (9 marks): (dy/dx)(3y/x+1)=(x+1)4 Q3- For the below initial value problem find y(2) (10 marks). y+y=1+cos2x,y(1)=4 Q4- verify that the indicated expression is an implicit solution of the given first-order differential equation (10 marks). dtdX=(X1)(12X);ln(X12X1)=t Q5- Verify whether the following differential equation is exact and find their general solutions (15 marks). (2xysec2x)dx+(x2+2y)dy=0 Q6- In the following problems find, the solution of the given initial-value ODEs (20 marks). dxdy=yex2,y(4)=1dxdy=y2sinx2,y(2)=31 Q7- Solve the following initial value problems ( 10 marks): a) 16y40y+25y=0, y(0)=3,y(0)=49 b) 4y24y+37y=0, y()=1,y()=0 Q8 - Use Euler's method with step size h=0.1 to approximate the solution to the initial value problem (10 marks): dxdy=4cos(x+y),y(0)=1atthepointsx=0,0.1,0.2,,1.0 Q9- Use Euler's method with step size h=0.2 to approximate the solution to the initial value problem (10 marks): dxdy=x1(y2+y),y(1)=1atthepointsx=1.2,1.4,1.6,1.8