Homoge 5. The equation a?y" + axy + by = 0 where a.b are constants is called Euler-Cauchy equation. Show that under the transfor- mation r = e' when r >0 the above reduces to y" + (a – 1)/ + by = 0 which is an equation with constant coefficients. Hence solve (i) ry" + 2ary – 12y = 0 (ii) a?y" + xy + y = 0 (iii) a?y" – xy + y = 0 6. Find differential equation with constant coefficients satisfied by the functions (ii) e-2" cos 3r, e-2r sin 3r, e-2x re-27. (i) a2, e", re" 7. Find the general solution of the following differential equations. d'y (a) + y(x) = 0. dri d³y (b) dr d'y dy = 0. dr dz4 By dy dy (c) dr ই + dr - y(x) = 0. d°y (d) d'y + 10 dr dy Ts + 10 +5 fip² °dz dr² + y(x) = 0. 8. Solve y" +y = tana in the interval (0, 7) using the method of variation of parameter. Answer: In | sect| +1- sin t(ln | sec t + tan t|). 9. Solve the following initial-value problems: (a) y" – 2y + y = 2re2" + 6e"; y(0) = 1, y'(0) = 0. (b) y"(x) + y(x) = 3x2 – 4 sin r, y(0) = 0, y'(0) = 1. 10. Use the method of undermined coefficients to find a particular solution to the following differential equations: (a) y" – 3y' + 2y = 2x² + 3e2r. (b) y"(r) – 3y (r)+ 2y(x) = xe²# + sin æ.

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.