Class 12 Differential Equations · Case Study4 Markslinear_differential_equations
Class 12 Maths · Differential Equations · Case Study #117
For a linear equation y′ + P(x)y = Q(x), an integrating factor is exp(∫P dx), and y·IF = ∫Q·IF dx + C. Consider y′ + y/x = x², x ≠ 0. Answer (i), (ii), and either (iii)(a) or (iii)(b).
Part (1) 1 Marks
Identify P(x).
Answer: P(x) = 1/x.
- Compare y′ + y/x = x² with y′ + Py = Q.
- The coefficient of y is P(x) = 1/x, x ≠ 0.
Part (2) 1 Marks
Identify Q(x).
Answer: Q(x) = x².
- In the normalized linear form, Q is the right-hand-side forcing function.
- Here Q(x) = x².
Part (3) 2 Marks
Find the general solution of y′ + y/x = x².
Answer: y = x³/4 + C/x, x ≠ 0.
- Use IF x on a connected interval x ≠ 0.
- Then (xy)′ = x³, so xy = x⁴/4 + C.
- Thus y = x³/4 + C/x, x ≠ 0.
Part (4) 2 Marks
Replace Q(x) by sin x and solve y′ + y/x = sin x.
Answer: y = −cos x + (sin x + C)/x, x ≠ 0.
- Use the same integrating factor x.
- Then (xy)′ = x sin x. By parts, xy = −x cos x + sin x + C.
- Thus y = −cos x + (sin x + C)/x, x ≠ 0.