NCERT Solutions For Class 12 Maths Chapter 9 Differential Equations Exercise 9.4 - 2025-26

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1. Solve the differential equation \[\left( {{x^2} + xy} \right)dy = \left( {{x^2} + {y^2}} \right)dy\]

Summary: This problem is about solving a homogeneous differential equation using substitution and integration.

• Start by rearranging the equation and recognizing it as homogeneous.
• Use the substitution \(y = vx\) and differentiate accordingly.
• After substituting and simplifying, separate variables for integration.
• Carry out the integration on both sides and then back-substitute the values.
• Final answer: \[ {\left( {x - y} \right)^2} = x{e^{\frac{y}{x}}}C \] where \(C = {e^{-C}}\).

2. Solve the differential equation \[y' = \dfrac{{x + y}}{x}\]

After rearranging: \(\dfrac{dy}{dx} = 1 + \frac{y}{x}\) This is a homogeneous equation. Substitute \(y = vx\), so \(\dfrac{dy}{dx} = v + x\dfrac{dv}{dx}\). Now, substitute and simplify: \[ v + x\dfrac{dv}{dx} = 1 + v \implies dv = \frac{1}{x}dx \] Integrating both sides: \[ v = \log x + C \] Back-substitute \(v = \frac{y}{x}\): \[ \frac{y}{x} = \log x + C \implies y = x\log x + Cx \]

3. Solve the differential equation \[\left( {x - y} \right)dy = \left( {x + y} \right)dx\]

Summary: A homogeneous equation is solved by substitution, breaking down the integration into smaller parts.

• Rearrange to get \(\frac{dy}{dx} = \frac{x + y}{x - y}\).
• Substitute \(y = vx\) and use \(\frac{dy}{dx} = v + x\frac{dv}{dx}\).
• After simplification and substituting, separate variables.
• Integration step involves splitting into easier integrals.
• Final answer: \[ \tan^{-1}\left(\frac{x}{y}\right) = \log\left(x^2 + y^2\right) + C \]

4. Solve the differential equation \[\left( {{x^2} - {y^2}} \right)dx + 2xydy = 0\]

Summary: Solve the homogeneous equation by substituting \(y = vx\), and integrating with substitution.

• Rearrange to get \(\frac{dy}{dx} = -\frac{x^2 - y^2}{2xy}\).
• Use the substitution \(y = vx\) and differentiate as needed.
• After substitution, separate the variables and integrate.
• Back-substitute to get the answer in terms of \(x\) and \(y\).
• Final answer: \[ x^2 + y^2 = Kx \] where \(K\) is a constant.

5. Solve the differential equation \[{x^2}\dfrac{{dy}}{{dx}} = {x^2} - 2{y^2} + xy\]

Summary: This equation is solved by recognizing homogeneity, making a substitution, and integrating.

• Rearrange to: \(\frac{dy}{dx} = \frac{x^2 - 2y^2 + xy}{x^2}\).
• Use \(y = vx\), and differentiate.
• Upon rearranging and simplifying, separate variables for integration.
• Use properties of integrals and trigonometric substitution where needed.
• Final answer: \[ \frac{1}{2\sqrt2}\log\left| \frac{x + \sqrt2 y}{x - \sqrt2 y} \right| = \log|x| + C \]

6. Solve the differential equation \[xdy - ydx = \sqrt {{x^2} + {y^2}} dx\]

Summary: Use substitution to solve a homogeneous equation with a square root term.

• Rearrange to: \(\dfrac{dy}{dx} = \dfrac{\sqrt{x^2 + y^2} + y}{x}\).
• Substitute \(y = vx\), differentiate, and substitute back.
• Simplify to separate variables.
• Integrate both sides, then express everything in terms of \(x\) and \(y\).
• Final answer: \[ y + \sqrt{x^2 + y^2} = Cx^2 \]

7. Solve the differential equation \[\left\{ {x\cos \left( {\dfrac{y}{x}} \right) + y\sin \left( {\dfrac{y}{x}} \right)} \right\}ydx = \left\{ {y\sin \left( {\dfrac{y}{x}} \right) - x\cos \left( {\dfrac{y}{x}} \right)} \right\}xdy\]

Summary: Substitution and careful integration turn a complex-looking equation into a solvable one.

• First, rearrange and note it's homogeneous.
• Substitute \(y = vx\), and differentiate.
• Set up the equation to separate variables before integration.
• Split the integrals as needed for easier calculation.
• Final answer: \[ y x \cos\left(\frac{y}{x}\right) = K \]

8. Solve the differential equation \[x\dfrac{{dy}}{{dx}} - y + x\sin \left( {\dfrac{y}{x}} \right) = 0\]

Summary: Use substitution and basic integration to solve this homogeneous differential equation.

• Rearrange to: \(\frac{dy}{dx} = \frac{y - x\sin(y/x)}{x}\).
• Substitute \(y = vx\) and differentiate.
• After simplifying, separate variables and integrate.
• Arrive at the answer after back-substituting and simplifying.
• Final answer: \[ x(1 - \cos(y/x)) = C\sin(y/x) \]

9. Solve the differential equation \[ydx + x\log \left( {\dfrac{y}{x}} \right)dy - 2xdy = 0\]

Summary: This question requires substitution and algebraic manipulation before integrating.

• Rearrange to get a homogeneous equation.
• Let \(y = vx\); then substitute and differentiate as usual.
• Simplify, separate variables, and integrate both sides.
• Use another substitution for ease: \(\log v - 1 = t\).
• Back-substitute to arrive at: \[ \log\left(\frac{y}{x}\right) - 1 = yC \]

10. Solve the differential equation \[\left( {1 + {e^{\dfrac{x}{y}}}} \right)dx + {e^{\dfrac{x}{y}}}\left( {1 - \dfrac{x}{y}} \right)dy = 0\]

Summary: The problem uses substitution and natural logarithms to find the solution.

• Rearrange and notice it's homogeneous.
• Use \(x = vy\) as the substitution; differentiate and substitute accordingly.
• Simplify, separate the variables, and perform the integration.
• Back-substitute \(v = x/y\) for the final answer.
• Final answer: \[ x + y e^{x/y} = C \]

11. Solve the differential equation \[\left( {x + y} \right)dy + \left( {x - y} \right)dx = 0;{\text{ }}y = 1;x = 1\]

Summary: Initial value problem solved by substituting, integrating, and then plugging values to find the constant.

• Rearrange for homogeneity.
• Substitute \(y = vx\), get differential equation, and separate variables.
• Integrate and use the given values \(y = 1\), \(x = 1\) to find \(C\).
• Final answer: \[ 2\tan^{-1}\left(\frac{y}{x}\right) + \log(x^2 + y^2) = \frac{\pi}{2} + \log 2 \]

12. Solve the differential equation \[{x^2}dy + \left( {xy + {y^2}} \right)dx = 0\]; \[y = 1\] when \[x = 1\].

Summary: After substituting, integrate and use the initial value to find the constant.

• Rearrange the equation and substitute \(y = vx\).
• Simplify, separate variables, and integrate both sides.
• Plug in \(y = 1, x = 1\) to solve for \(C\).
• Final answer: \[ 2x + y = 3y x^2 \]

13. Solve the differential equation \[\left[ {x{{\sin }^2}\left( {\dfrac{y}{x}} \right) - y} \right]dx + xdy = 0\]; \[y = \dfrac{\pi }{4}\] when \[x = 1\].

Summary: Substitute and integrate; use initial value to find the particular solution.

• Rearrange and substitute \(y = vx\), then differentiate.
• Simplify, separate variables, and integrate.
• Plug in \(y = \frac{\pi}{4}, x = 1\) to find \(C\).
• Final answer: \[ \cot\left(\frac{y}{x}\right) = \log|xe| \]

14. Solve the differential equation \[\dfrac{{dy}}{{dx}} - \dfrac{y}{x} + \cos ec\left( {\dfrac{y}{x}} \right) = 0\]; \[y = 0\] when \[x = 1\].

Summary: Substitution, integration, and initial values lead to the required solution.

• Write as a homogeneous equation and use \(y = vx\).
• After substitution, separate variables and integrate both sides.
• Plug \(y = 0\) and \(x = 1\) to determine the integration constant.
• Final answer: \[ \cos\left(\frac{y}{x}\right) = \log|xe| \]

15. Solve the differential equation \[2xy + {y^2} - 2{x^2}\dfrac{{dy}}{{dx}} = 0\]; \[y = 2\] when \[x = 1\].

Summary: Substitution and integration then plugging initial values gives the specific solution.

• Rearrange for homogeneity, then substitute \(y = vx\).
• Separate variables after differentiating; integrate both sides.
• Plug \(y = 2, x = 1\) to compute \(C\).
• Final answer: \[ y = \frac{2x}{1 - \log x} \quad (x \neq e) \]

16. A homogeneous differential equation of the form \[\dfrac{{dx}}{{dy}} = h\left( {\dfrac{x}{y}} \right)\] can be solved by making the substitution. \[ \left( A \right)y = vx{\text{                      }}\left( B \right)v = yx\]   \[\left( C \right)x = vy{\text{                      }}\left( D \right)x = v\]

As \(h\left(\frac{x}{y}\right)\) is a function of \(\frac{x}{y}\), we need to use the substitution \(x = vy\). So, the correct option is (C).

17. Which of the following is a homogeneous differential equation?
(A) \((4x + 6y + 5)dy - (3y + 2x + 4)dx = 0\)
(B) \((xy)dx - (x^3 + y^3)dy = 0\)
(C) \((x^3 + 2y^2)dx + 2xydy = 0\)
(D) \(y^2 dx + (x^2 - xy - y^2)dy = 0\)

The correct option is (D).
Explanation:
Rewrite the equation as: \[ \frac{dx}{dy} = -\frac{x^2 - xy - y^2}{y^2} \] If you put \(x = kx\), \(y = ky\), all terms remain the same degree showing homogeneity. So, the equation in option (D) is homogeneous.

What You’ll Practice in Differential Equations Class 12 Exercise 9.4

• Learn the substitution y = vx to solve homogeneous differential equations in exercise 9.4 class 12 maths.
• Practice separating variables before integrating differential equations for clear solutions.
• Get step-by-step experience in handling both basic and challenging integrals.
• See how to check if an equation is homogeneous using degrees of terms.
• Understand why each step—substitution, rearrangement, integration, and constant—is important.

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